thomas-szymczak.dethomas-szymczak.de/dissertation.pdf · Vorwort Die hoc¨ hste Klugheit besteht...

Domination Problems on Special Graph Classes

Dissertation

zurErlangung des akademischen Grades

Doktor-Ingenieur (Dr.-Ing.)

der Fakultat fur Ingenieurwissenschaften

der Universitat Rostock

vorgelegt von

Thomas Szymczak, geb. am 13.03.1971 in Dinslaken (NRW)

aus Rostock

Rostock, 07.12.2001

Datum der Verteidigung: 18. Oktober 2002

Gutachter

Prof. Dr. Andreas BrandstadtUniversitat RostockFachbereich Informatik18051 RostockDeutschland

Prof. Dr. Dieter KratschUniversite de MetzUFR MIMDepartement d’informatiqueIle du Saulcy57045 Metz Cedex 1France

Prof. Dr. Feodor F. DraganKent State UniversityDepartment of Computer ScienceKent, Ohio 44242U.S.A.

Vorwort

Die hochste Klugheit besteht darin,den Wert der Dinge genau zu kennen.(Francois Duc de La Rochefoucauld)

Viele Probleme, die sich mit dem strategischen Plazieren von irgendwelchen Objekten in ei-nem Netzwerk beschaftigen, lassen sich als Dominationsprobleme in gewissen Graphen be-schreiben. Die Wurzeln von Domination gehen bis in das Jahr 1850 zuruck, wo sich NAUCK,GAUSS ET AL. mit dem Plazieren von Schachfiguren auf einem �� –Brett beschaftigten.Ziel war es, moglichst wenig Figuren so auf dem Brett zu plazieren, daß alle Felder dominiertwerden.

1975 wurde dann mit der algorithmischen Untersuchung des Problems MINIMUM DO-MINATING SET begonnen. JOHNSON war der erste, der die

��–Vollstandigkeit dieses Pro-

blems fur allgemeine Graphen erkannte; COCKAYNE ET AL. entwickelten den ersten Linear-zeitalgorithmus fur die Klasse der Baume. Seitdem sind unzahlige Arbeiten uber Dominationentstanden.

1995 habe ich bei der Erstellung meiner Diplomarbeit den Zugang zu Domination ge-funden. Mein Interesse galt insbesondere der Entwicklung von effizienten Algorithmen furmoglichst große Graphenklassen. Diese Arbeit setzte ich wahrend meiner Tatigkeit als Wis-senschaftlicher Assistent am Institut fur Theoretische Informatik an der Universiat Rostockfort, wo auch diese Abhandlung entstand.

Die Resultate stammen vorwiegend aus folgenden Arbeiten:� A. BRANDSTADT, T. KLEMBT, V.B. LE, S. MAHFUD, T. SZYMCZAK, On the struc-

ture and clique width of graph classes defined by two forbidden four-vertex graphs,Manuskript, 2001.

� F. NICOLAI, T. SZYMCZAK, � –Domination problems on trees and their homogeneousextensions, International Journal of Mathematical Algorithms 1 (1999), 53–79.

� F. NICOLAI, T. SZYMCZAK, Homogeneous sets and domination: A linear time algo-rithm for distance–hereditary graphs, Networks 37(3) (2001), 117–128.

Neben meinen Coautoren danke ich besonders PD Dr. Van Bang Le und Prof. Dr. AndreasBrandstadt fur die wertvollen Korrekturhinweise und die vielen fruchtbaren Diskussionen.

Rostock, im Dezember 2001 Thomas Szymczak

III

IV VORWORT

Contents

1 Introduction 1

2 Preliminaries 32.1 Domination problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Relationship between domination problems . . . . . . . . . . . . . 6

3 Special Graph Classes 93.1 Classes of bounded clique width . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Definition and some properties . . . . . . . . . . . . . . . . . . . . 10

3.1.2 Monadic second–order logic and the class LinEMSOL( �� ) . . . . 11

3.1.3 Graph classes defined by two forbidden four–vertex graphs . . . . 15

3.1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Graph classes defined by small forbidden subgraphs . . . . . . . . . . . . . 20

3.2.1�

–free graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 � � �� –free graphs . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 The class �� — contraction of hom. sets . . . . . . . . . . . . 29

3.3.1 Homogeneous Extensions of Graphs . . . . . . . . . . . . . . . . . 29

3.3.2 The � –domination set problem on homogeneous extensions of trees 35

3.3.3 The generalized � –domination set problem and homogeneous sets . 38

3.3.4 The � –dominating set problem on �� !�� . . . . . . . . . 45

3.3.5 The � –dominating set problem on distance–hereditary graphs . . . 48

3.3.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 48

A Complexity of Domination Problems 51A.1 Abbreviations for problems . . . . . . . . . . . . . . . . . . . . . . . . . . 53

A.2 Complexity on graph classes . . . . . . . . . . . . . . . . . . . . . . . . . 53

A.2.1 1–CUBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

A.2.2 2–CUBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

A.2.3 AT–free graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

V

VI CONTENTS

A.2.4 bipartite graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A.2.5 chordal graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A.2.6 chordal bipartite graphs . . . . . . . . . . . . . . . . . . . . . . . . 55

A.2.7 circle graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A.2.8 circular–arc graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A.2.9 claw–free AT–free graphs . . . . . . . . . . . . . . . . . . . . . . 57

A.2.10 co–comparability graphs . . . . . . . . . . . . . . . . . . . . . . . 57

A.2.11 comparability graphs . . . . . . . . . . . . . . . . . . . . . . . . . 57

A.2.12 convex bipartite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A.2.13 convex–round graphs . . . . . . . . . . . . . . . . . . . . . . . . . 58

A.2.14 distance–hereditary graphs . . . . . . . . . . . . . . . . . . . . . . 58

A.2.15 DSP–graphs (graphs having a dominating shortest path) . . . . . . 59

A.2.16 doubly chordal graphs . . . . . . . . . . . . . . . . . . . . . . . . 59

A.2.17 dually chordal graphs . . . . . . . . . . . . . . . . . . . . . . . . . 60

A.2.18 graph classes of bounded clique width . . . . . . . . . . . . . . . . 60

A.2.19 homogeneously orderable graphs . . . . . . . . . . . . . . . . . . 61

A.2.20 interval graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

A.2.21 line graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

A.2.22 partial � –trees (for bounded � ) . . . . . . . . . . . . . . . . . . . . 62

A.2.23 permutation graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 62

A.2.24 planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

A.2.25 planar bipartite graphs . . . . . . . . . . . . . . . . . . . . . . . . 63

A.2.26 � –polygon graphs for fixed �� . . . . . . . . . . . . . . . . . . 64

A.2.27 series–parallel graphs = partial 2–trees . . . . . . . . . . . . . . . . 64

A.2.28 split graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

A.2.29 strongly chordal . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

A.2.30 trapezoid graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A.2.31 trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A.2.32 undirected path graphs . . . . . . . . . . . . . . . . . . . . . . . . 67

A.2.33 weakly chordal graphs . . . . . . . . . . . . . . . . . . . . . . . . 67

Index 79

Thesen 83

Tabellarischer Lebenslauf 85

Erklarungen gemaß � 3, Absatz 1, Punkt 7 und 8 der Promotionsordnung 87

List of Figures

3.1 A graph of clique width three. . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Clique width of �� -free graphs. . . . . . . . . . . . . . . . . . . . . 16

3.3 The graph �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Graphs containing exactly two nodes. . . . . . . . . . . . . . . . . . . . . 23

3.5 Graphs containing exactly three nodes. . . . . . . . . . . . . . . . . . . . 23

3.6 Graphs containing exactly four nodes. . . . . . . . . . . . . . . . . . . . . 23

3.7 Hanging of the edge �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.8 MINIMUM DOMINATING SET on �� -free graphs. . . . . . . . . . . . 28

3.9 Example for the operations �� , �� and �� . . . . . . . . . . . . . 31

3.10 Generation of a graph in �� which is not in � �� trees�. . . 32

3.11 Inclusion hierarchy of some graph classes. . . . . . . . . . . . . . . . . . 33

3.12 An example for the reduction of maximal homogeneous sets to vertices ormeta–vertices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.13 An example to the algorithm presented in [9]. . . . . . . . . . . . . . . . . 37

3.14 The problem with meta–vertices. . . . . . . . . . . . . . . . . . . . . . . . 38

3.15 An example for Case 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.16 An example for the proof of Theorem 3.3.12. . . . . . . . . . . . . . . . . 47

A.1 The complexity of the minimum dominating set problem on some selectedgraph classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

VII

VIII LIST OF FIGURES

List of Tables

3.1 Some graph classes with bounded an unbounded clique width. . . . . . . . 20

3.2 Complexity of the Edge Domination Problem on special graph classes. . . . 27

IX

X LIST OF TABLES

Chapter 1

Introduction

Location problems play an important role in network design. Let � be a network structureand let processors of two types (suppliers and receivers, for modelling a coverage problem intelecommunication) be assigned to the vertices of � . To each receiver a value is associatedindicating the radius within which it can receive information from suppliers. Assuming thatthe production effort for suppliers is much higher than for receivers we have to minimizethe number of suppliers which are necessary to provide all receivers with information. If allreceivers are of the same type, i.e. their radius value is identical, then this is exactly the well–known � –domination problem on graphs (cf. [42]): Compute a minimum cardinality set �such that for each vertex � outside � there is at least one vertex inside � of distance at most� to � . Assigning a supplier to each vertex of � then minimizes their number. If we allowdifferent types of receivers, i.e. the radii do not coincide, then we have the more general� –domination problem: Given a graph � and a radius function ��

compute aminimum cardinality set � such that for each vertex � outside � there is at least one vertexinside � of distance at most � �� . Note that � �� means that this vertex must belong to� . So we can extend a given substructure of the network by assigning radius zero to alreadyinstalled suppliers.

It is well–known that the domination problem is� �

–complete for general graphs (JOHN-SON, circa 1975). In recent years the behaviour of certain graph classes with respect to (sev-eral) domination problems were investigated. Hereby, the following two approaches play animportant role for solving domination problems on a graph class :

� The design of a tree representation for . Very often efficient algorithms for domina-tion can be developed by using an underlying tree structure of a graph (cf. [9, 47, 55]).In [47] the authors review the complexity of the minimum dominating set problem onseveral families of perfect graphs using their tree representations.

� Shrinking homogeneous sets to smaller components and then, recursively solve theproblem (cf. [54, 108]).

A combination of both approaches was used in [108] for solving the � –dominating set prob-lem on homogeneous extensions of trees in linear time. Hereby, a graph is a homogeneousextension of a tree iff the reduction of all homogeneous sets to single vertices gives a tree.

1

2 CHAPTER 1. INTRODUCTION

More general, we obtained in [108], that efficient algorithms solving the � –dominating cliqueand the connected � –dominating set problem on a hereditary graph class lead to efficientalgorithms on their homogeneous extensions, too.

In this thesis we extend the results of [108] regarding the � –dominating set problem to asuperclass of homogeneous extensions, namely �� .

Recently, the concept of clique width attracted much attention. In [40] the authors givean unified approach to get efficient solutions for many algorithmic graph problems on graphclasses of bounded clique width. One condition needed for this is that the given problemis expressible in terms of a logical expression in a so–called Monadic Second Order Logic.MINIMUM DOMINATING SET and most of the variants of this problem are such problemstherefore bounded clique width is an important method solving domination problems.

Many hereditary graph classes can be characterized by forbidden induced subgraphs or,if not, at least a list of small forbidden subgraphs is known (see [23] for an overview of suchclasses). Thus, it is interesting to consider � –free graphs where � is a set of graphs contain-ing at most four vertices. For all of these classes we investigate if MINIMUM DOMINATING

SET is��

–complete or if it can be solved by a polynomial time algorithm.

The paper is organized as follows.

At first we give basic definitions and notions (chapter 2).

After that in chapter 3 we work on domination problems on special graph classes. Ini-tially, we consider graph classes of bounded clique width. We investigate which dominationproblems can be expressed in Monadic Second Order Logic (see Theorem 3.1.3) and provethat the clique width of ( � � ,co-paw)-free graphs (resp. ( � � ,diamond, � � ,claw)–free graphs)is bounded (resp. unbounded). In [20] we extend this to nearly all combinations of graphclasses defined by exactly two forbidden four–vertex graphs.

Using these results we consider � –free graphs where � is a set of graphs containing atmost four vertices. For all of these classes we investigate if MINIMUM DOMINATING SET is��

–complete or if it can be solved by a polynomial time algorithm (see Corollary 3.2.14).

Next, we consider the relation of homogeneous sets to the minimum � –dominating setproblem. We show that one can reduce a homogeneous set to one vertex or to a so–calledmeta–vertex consisting of two nonadjacent vertices with � –value one. This reduction to-gether with modular decomposition leads to an � �� time algorithm for computing aminimum � –dominating set for graphs which can be generated from the one–vertex graphby a finite number of homogeneous extensions (substitution of an arbitrary graph into a ver-tex) and by attaching pendant vertices (leaves). For distance–hereditary graphs — a propersubclass of this graph class — we even get a linear time algorithm.

In the appendix we list the complexity of some domination problems on differentgraph classes (but we declare no claim to completeness). This information can bea starting point to extend the information system on graph class inclusions (ISGCI,http://www.informatik.uni-rostock.de/˜gdb/isgci/Isgci.html)with information about the complexity of important graph theoretic problems.

Chapter 2

Preliminaries

Throughout this paper all graphs are finite, undirected and simple (i.e. loop–free and withoutmultiple edges).

Let � � � �� be a graph with vertex set � � � ��

and edge set � � � �� . If no

confusion can arise we write � � � � � and � � � �� .For an edge

�� we write shortly� � . A vertex

�is called a neighbor of � iff

� �� .The set �� of all neighbors of � is called the neighborhood of � . For a subset �� wedefine � � � ��

. Further, we write �� and �� .

A graph �� is called a subgraph of � iff �� and �� . �� is aninduced subgraph iff � � is a subgraph of � and � � ! � � � �#" � .1 We also say that � � isinduced by �� and usually write � � �� for �� . If �� $�$�$ � �&% � is a � –tuple of verticesof � we analogously define � � �� $�$�$ � �&%�� .

Let � be a vertex in � . For the graph � � �(' � �� we write shortly �*) � .The degree of a vertex �+� � , i.e. the number of neighbors of � in � , is denoted by

� �-,/. �� , the maximum degree of a vertex in � by 0 �� .

A sequence 1 �� $�$�$ � �&% � of pairwise distinct vertices is a path in � iff for all 23��54 ��$�$�$ � �6) 4 � holds � � � �87 �9� � . The length of 1 is ��) 4 . 1 is chordless iff � � �� :1 � � � ��) 4 .

A path � �� $�$�$ � �&% � is a cycle in � iff ��;(< and � � �&%�� . The length of � is � . �is chordless iff � � �� .

The distance � �>= �@? � between two vertices = , ?A� � is the minimum length of a pathbetween

�and � , or B if there is no such path. For �C� � and D� � we define � �� E � �

� �GF �� .

The � –th neighborhood % �� of a vertex � of � is the set of all vertices of distance � to� , i.e.

% �� . � � � � � �H� �

1 I9J:KGLNM denotes the set of all O –element subsets of L .

3

4 CHAPTER 2. PRELIMINARIES

whereas the disk of radius � centered at � is the set of all vertices of distance at most � to � :

� �� . � � � � �� H� %�� $

The eccentricity � �� of a vertex � � � is the maximum of � �� = �taken over all = � � .

The maximum over the eccentricities of all vertices of � is the diameter � �� of � .

� is connected iff for all�

, � � � there is a path in � connecting�

and � , otherwise wecall � disconnected.

Let �� be a subset of � .

� �� is a connected component in � iff � � �3� � is connected and for all =N� � ' �3� holds� � �� = � � is not connected.

� � � is a clique (complete set) in � iff for all�

, � � � ,�� holds

� � � � .

� � � is a stable set (independent set) in � iff for all�

, � � � ,�� holds

� �� .

� is complete iff � �� is a clique in � , � is edgeless iff � ��

is a stable set in � .

A set� � � is called homogeneous set iff any pair of vertices of

�has the same

neighborhood outside�

:

� � � " � �(' � � �� " � �(' � �for all

� � � � � $A homogeneous set

�is trivial (resp. nontrivial) iff � � �5� � � � 4 � � �� (resp.

4�� ).A graph containing only trivial homogeneous sets is called prime.

Let � � � �� be a tree, i.e. a connected cycle–free graph, rooted at � and let � some

vertex of � . We denote by �� the subtree of � rooted at � , i.e. �� contains any vertex�

suchthat � lies on the (unique) path connecting

�and the root � .

Let � be a set of graphs. A graph � is called � -free iff for all � � � holds: � is notan induced subgraph of � . If � � � ��$�$�$ �� (resp. � � �� ) is finite (resp. has exactlyone element) we write �� $�$�$ �� –free (resp. � –free) instead of � -free.

Let � be a class of graphs. � is called hereditary iff for every � �� and � � �� holds: � � � �� . Graph classes defined by forbidden subgraphs are examples for hereditarygraph classes.

Let � , � be subsets of � . We write � �"!#� iff for all $ �� , % �� holds $&% � � . Wewrite �'��()� iff for all $ �*� , % �+� holds $&%, � � .

For vertex–disjoint graphs � � � �� and �3� � �� we define

� ��-( �� , the union of � and �� (in the literature often the unionis denoted by � � �� or ��. �� ),

� � � � ��( $�$�$/��( �0 132 45 times

, � � 4 an arbitrary integer,

� ��6! �� =H? � = � � �@?C� �� , the join of � and �� .

2.1. DOMINATION PROBLEMS 5

Some special graphs get a standard notation ( � � � ):

� 1�� := graph of a chordless path on � vertices,

� �� := graph of a chordless cycle on � vertices ( � ;(< ),� �� := complete graph on � vertices.

By the above notion � � � is an edgeless graph on � vertices.

An induced connected subgraph�

of � is an isometric subgraph of � iff the distanceof any two vertices = �@? in

�equals their distance in � , i.e. �� >= �@? � � . �>= �@? � for all= �@?C� � � � �

. Then, a connected graph � is distance–hereditary iff every induced connectedsubgraph of � is isometric ([69]).

In the 1980s certain characterizations of distance–hereditary graphs were given. A con-structive generation of distance–hereditary graphs was presented in [24] via so–called onevertex extensions. In [77], analogous results were obtained including a linear time recogni-tion algorithm which constructs a sequence of one vertex extensions.

Let �� be a graph, =H� �� and = �� . We extend the graph �� to � byadding vertex = and joining it to

� only =�� — the pendant vertex operation PV,

� all neighbors of =H� — the false twin operation FT,

� =�� and all its neighbors — the true twin operation TT.

Then, a connected graph � with at least two vertices is distance–hereditary if and only if� can be obtained from an edge by a sequence of one vertex extensions PV, FT and TT.

2.1 Domination problems

The roots of domination lie in 1850 when NAUCK, GAUSS ET AL. studied the placementof chess pieces on a � � � –board. They investigated the domination of all squares by aminimum number of pieces.

Later, the problem was formulated for graphs. A set � � � is a dominating set in �if every vertex � � �*' � has at least one neighbor in � . A dominating set � is minimumif there is no dominating set � � with � � � � � � � � . The cardinality of a minimum dominat-ing set in � is called domination number and denoted by ��

. The problem MINIMUM

DOMINATING SET consists of computing a minimum dominating set for a given graph.

In the last years many variants of MINIMUM DOMINATING SET have been developed.Here, we want to mention four well–known general modifications:

� Additionally, we have given a radius function � � � � �. � is a � -dominating set if

for every vertex �� (' � there is at least one vertex �C� � of distance at most � �� to � . If � �� 4

for all � � � we get the classical definition for dominating set. Note,that � �� means that this vertex must belong to every � -dominating set.


� � is called a perfect dominating set if every vertex � � � ' � has exactly one neighborin � .

� We study dominating sets � with additional properties:

– � � � �is connected� MINIMUM CONNECTED DOMINATING SET,

– � is an independent set in �� MINIMUM INDEPENDENT DOMINATING SET,

– � is total in � i.e. � � � �contains no isolated vertices (= vertices of degree 0)� MINIMUM TOTAL DOMINATING SET,

– � is a clique� MINIMUM DOMINATING CLIQUE.

� We consider (vertex) weighted versions. Let � � � ��be a function assigning

weights to vertices of � . A dominating set � is a minimum weighted dominating set if� is a dominating set in � and there is no dominating set � � in � with �� . If �� 4

for all � � � the weighted version is the classical dominationproblem.

These modifications can also be combined. For example: � is a connected perfect dominat-ing set if � is a perfect dominating set and � � � �

is connected.

Finally, we want to mention the problem (CARDINALITY) STEINER TREE. Let �� be a subset of vertices of � . � is called a Steiner set if � is a vertex set containing � suchthat � � � �

is connected. STEINER TREE consists of computing a Steiner set of minimumcardinality. This problem has a lot of applications in VLSI design and reconstruction ofphylogenetic trees in biology.

2.1.1 Relationship between domination problems

In [124] the authors investigate the relationship between Steiner trees and connected domi-nating sets in chordal graphs. Using a convexity property they obtain

Theorem 2.1.1 ([124]) The connected dominating set problem is polynomial for any classof chordal graphs for which the cardinality Steiner tree problem is polynomial. Moreover,the cardinality Steiner tree problem is

��–complete for any subclass of chordal graphs for

which the connected dominating set problem is��

–complete.

In [92] the authors investigate the relationship between MINIMUM DOMINATING SET andMINIMUM TOTAL DOMINATING SET. The work was motivated by the fact that in almostall of the known cases the two problems have the same complexity status. Looking in theAppendix one can find one exception: MINIMUM DOMINATING SET is

��–complete on

chordal–bipartite graphs whereas MINIMUM TOTAL DOMINATING SET on this class can besolved in polynomial time.

2.1. DOMINATION PROBLEMS 7

For a graph � � � �� , � � � � ��$�$�$ � � �/� , the duplex graph � ��

has vertices � � � � �� $�$�$ � � �5� and edges � � � � �� .Theorem 2.1.2 ([92]) Let be a graph class with � � � � . Then holds:

1. If there is an algorithm solving MINIMUM DOMINATING SET in time � �� then

there is also an algorithm solving MINIMUM TOTAL DOMINATING SET in the sametime bound.

2. If MINIMUM TOTAL DOMINATING SET is��

–complete on then MINIMUM DOM-INATING SET is

��–complete on , too.

Since graph classes closed under adding false twins fulfill the property � � � �� of Theo-rem 2.1.2 ([92]) we can use this theorem for the following classes:

AT–free graphs, bipartite graphs, chordal bipartite graphs, circle graphs, co–comparability graphs, comparability graphs, convex bipartite graphs, distance–hereditary graphs, dually chordal graphs, homogeneously orderable graphs, per-mutation graphs, � –polygon graphs.

Chapter 3

Special Graph Classes

Since nearly all domination problems are��

–complete on general graphs it is interesting toinvestigate the complexity on special graph classes.

In the first section we consider graph classes of bounded clique width. For such classesunder certain conditions there exists a unified approach to get efficient solutions for manygraph theoretic problems. One condition is that the given problem can be expressed in termsof a logical expression in a so–called Monadic Second Order Logic. In particular we inves-tigate which domination problems can be expressed in such a logic.

In section two we look at � –free graphs where � is a set of graphs containing at mostfour vertices. For all of these classes we investigate if MINIMUM DOMINATING SET is

��–

complete or if it can be solved efficiently. For the case that only one graph is forbidden(i.e.

�–free graphs) we give the complexity status for MINIMUM DOMINATING SET and

MINIMUM CONNECTED DOMINATING SET for arbitrary graphs�

(i.e. not only for thecase � � � � � � �� ).

Next, we consider the relation of homogeneous sets to the minimum � –dominating setproblem. We show that one can reduce a homogeneous set to one vertex or to a so–calledmeta–vertex consisting of two nonadjacent vertices with � –value one. This reduction to-gether with modular decomposition leads to an � �� time algorithm for computing aminimum � –dominating set for graphs which can be generated from the one–vertex graphby a finite number of homogeneous extensions (substitution of an arbitrary graph into a ver-tex) and by attaching pendant vertices (leaves). For distance–hereditary graphs — a propersubclass of this graph class — we even get a linear time algorithm.

3.1 Classes of bounded clique width

Graphs of clique width � , � � � , were introduced by COURCELLE, ENGELFRIET andROZENBERG (1993) in [32] as graphs which can be defined by � –expressions based ongraph operations which use � vertex labels. In the first subsection we give the exact defini-tion for clique width and some properties with respect to modular decomposition and graphcomplement.

9

10 CHAPTER 3. SPECIAL GRAPH CLASSES

Recently, the concept of clique width attracted much attention. In [40] the authors give anunified approach to get efficient solutions for many graph theoretic problems on graph classesof bounded clique width provided a � –expression is given. For this, the given problem mustbe expressible in terms of a logical expression in a so called Monadic Second Order Logic.This concept will be outlined in the second subsection. In particular, we investigate whichdomination problems can be expressed in Monadic Second Order Logic (see Theorem 3.1.3,Observation 3.1.4).

In [20] we investigate the structure and clique width of graph classes defined by exactlytwo forbidden four–vertex graphs. For nearly all combinations we prove that the clique widthis bounded, or not; we list our results in the third subsection. To give the reader a feeling howto prove that a graph class has bounded or unbounded clique width we give one example forboth cases.

Finally, we give an overview about the clique width of some important graph classes.

3.1.1 Definition and some properties

A � –graph, �� , is a graph with vertex labels in�54 ��$�$�$ � � � . Hereby, we force that every

vertex has exactly one label. If � is a � –graph we denote with � � �� the set of vertices with

label 2 , 2 4 ��$�$�$ � � .We consider the following graph operations:

� (Generation of an one vertex � –graph)Let � be an object and

� � �54 ��$�$�$ � � � . Then� �� denotes the � –graph � � � �� ,

� � �� .� (Disjoint union)

Let � and�

be � –graphs. Then, �� denotes the disjoint union of � and

�with

vertex labels �� , 2 4 ��$�$�$ � � .

� (Join between vertices of different labels)Let � be a � –graph. � � � ��

denotes the graph that we obtain from � if we add alledges between a vertex of label 2 and a vertex of label � and not changing any label.

� (Relabeling vertices of one label to another label)Let � be a � –graph. ��

denote the graph that we obtain from � if we change thelabel of all vertices with label 2 to � , i.e. ��

, � �� .

The clique width of a graph � , denoted by � � �� , is the minimum number of labels needed

to generate � by using these four operations. The recursive generation sequence of building� using the above operations by using only � different labels is called a � –expression (seeFigure 3.1 for an example).

If one want to show that a hereditary graph class � has bounded clique width it is enoughto prove this property for all prime graphs contained in � . Further, the property is invariant

3.1. CLASSES OF BOUNDED CLIQUE WIDTH 11

�$

��

��

% �

��

�� 2

��

�

� � � �� < �� 4 �� 2 � � �A< � � � � 4 � � � � � � � �� . 4 �� 4 ��% � �A< ��$ � � � � � � � � �� $

Figure 3.1: A graph of clique width three.

for building complements, i.e. � has bounded clique width if and only if

co– � � ��

has bounded clique width.

Theorem 3.1.1 ([40, 43]) The clique width of a graph is the maximum of the clique widthof its induced prime subgraphs, and the clique width of the complement graph � of � is atmost twice the clique width of � .

3.1.2 Monadic second–order logic and the class LinEMSOL( � ! � � )First, we define first–order logic (FOL) where quantification is allowed only over variables.Let � � � = � � 2 � � � and � ��

� � 2 � � � � � . We call the elements of � variablesand the elements of � relation symbols. Each relation symbol

� �� has an arity, denoted by

�� , which is equal to the upper index � .

A formula in first–order logic is defined recursively as follows:

1. For� �� , �

�� and variables = � ��$�$�$ � = � let

� �>=�� $�$�$ � = � � be a formula. Thesespecial formulas we call atomic formulas.

2. Let � , � be formulas and = be a variable. Then �� , �� , � � , � = � , �/= � areformulas.

3. There are no more formulas.

Next, we have to give the semantics of our formulas. A structure for a formula � in first–order logic is a tuple � � �� @2�� consisting of a non–empty set �� , the domain of � ,and an interpretation 2�� that assigns to every

� � � (if�

occurs in � ) a �� –ary relation� � � 2��

on �!� and to every =D�"� (if = occurs in � and = is in � unbounded bya quantor) an element = � � 2�� >= �

in �!� . Finally, we have to assign to � a logical value� �� 4 � under the structure � . This is done recursively as follows:


1. If � � �>=�� $�$�$ � = � � then � �� 4if and only if � �>= � � � ��$�$�$ � �>= � � � � � � � .

2. If � �� then � �� 4 ) � �� .

3. If � � � �then � �� GF � � ��

.

4. If � � � �then � ��

.

5. If � � = � then � �� GF � �� .

6. If � �/= � then � �� .

Hereby, �� is the extension of � by =� � . So, first–order logic coincides with the usual

first order predicate logic without function symbols (see for instance [115]).

Next, second–order logic (SOL) is an extension of first–order logic in terms of that quan-tification is allowed over relation symbols, too. Syntax and semantics are modified naturally— we omit the details.

Now, we are able to say what monadic second–order logic (MSOL) is. In this specialkind of second–order logic quantification is allowed only over variables and relation symbolsof arity one. If

�is a relation symbol of arity one and = is a variable then we write = � �

instead of� �>= �

.

For expressing properties of (labeled) graphs we use special formulas and structures. Let� � � FOL � SOL � MSOL � and � be an integer. With � �� we denote the set of all formulas �in logic

�that fulfill the following conditions:

� � contains only the following free relation symbols: � ��, � � , � � ��$�$�$ �� .

� � ��

and � � have arity two. We write �>= �@? � � � � � (resp. = ? ) for � ��>= �@? � (resp.

� � �>= �@? � ).� � � ��$�$�$ �� are unary.

Let � � � �� be a � –graph and � � � �� . We define the structure � .�� as follows:

� � �� ,

� for all � , � � � holds �� +� � � � �� if and only if �&�� ,1

� for all � , � � � holds �� +� � �� if and only if � � ,

� for all �C� � and 2 � �54 ��$�$�$ � � � holds � � �� if and only if vertex � has label 2 .

We use � �� .�� a � –graph � � for expressing properties of labeled graphs. Theindex ”

4” denotes that this is one possibility — see [31, 40] for other variants.

Let � be a property for graphs. We say that a formula � � � �� expresses � if thefollowing property is fulfilled:

1Note, that we only consider undirected graphs.


For all � –graphs � holds: � .�� 4if and only if � fulfills property � .

In the following we give some examples for such expressions. We use the following abbre-viations:

= � ? � �#= � ? � = ? � � �>= ? � $Theorem 3.1.2 The following graph properties can be expressed by a formula in �� :

1. � is complete,

2. � is edgeless,

3. � is total, i.e. � has no isolated vertices,

4. � is connected,

5. � is cycle–free.

Proof. The following formulas express the given properties:

1. � � � = � ? �>= ? � � �>= �@? � �+� � � ,2. � � = � ? �>= ? � � � �>= �@? � �+� � � ,3. � � � = � ? �>= ? � � �>= �@? � �+� � � ,4. � � �� N� � � �� C�� +� � � � � � �� :� � � � =�= ��

,

5. We use the following characterization: A graph � is cycle–free iff there is no vertex= � � such that two different neighbors ? , � � �>= �of = are in the same connected

component in �*)�= .

�� = � ? �� >= �@? � �+� � � � �>= �� +� � � � ? � �� H � =N� �� ?C� �

� � � � � � � � ��= � � � � � � � �+� � � � � �C� ��#$ �Analogously we are able to express optimization problems. We call an optimization problem� a LinEMSOL( � �� ) problem if and only if for every instance � of � the following propertiesare fulfilled:

� there is a MSOL–formula � � � �� $�$�$ �� where �� $�$�$ �� are free rela-tion symbols of arity one and it holds: �6�� $�$�$ � � � � is a solution for ��

iff� .�� 4

,


� there are evaluation functions � � ��$�$�$ � �� associating integer values to the vertices of� , integers $ � � � , 4 � 2 � � , 4 � � � � and �� GF � � � �H� such that the followingvalue is the optimum solution for the instance � of � :

��

�� $ � �

�� $ � � �6� ��$�$�$ � � � � is a solution for �� $Theorem 3.1.3 The following optimization problems are for every fixed integer � inLinEMSOL( � �� ):

1. MINIMUM WEIGHTED ( � –)DOMINATING SET,

2. MINIMUM WEIGHTED PERFECT DOMINATING SET,

3. MINIMUM WEIGHTED CONNECTED ( � –)DOMINATING SET,

4. MINIMUM WEIGHTED CONNECTED PERFECT DOMINATING SET,

5. MINIMUM WEIGHTED INDEPENDENT ( � –)DOMINATING SET,

6. MINIMUM WEIGHTED INDEPENDENT PERFECT DOMINATING SET,

7. MINIMUM WEIGHTED TOTAL ( � –)DOMINATING SET,

8. MINIMUM WEIGHTED TOTAL PERFECT DOMINATING SET,

9. MINIMUM WEIGHTED ( � –)DOMINATING CLIQUE,

10. STEINER TREE.

Proof. The following formula expresses that a set is a dominating set in � :

� �� =�= �� ? � ?C�� >= �@? � �+� � � � $It is easy to see that for fixed � one can also express that a set is a � –dominating set.Therefore, with

� � $ �� 4and �� the given function associating weights to the

vertices one can see that MINIMUM WEIGHTED ( � –)DOMINATING SET is an element inLinEMSOL( � �� ).

The following formula expresses that a set is a perfect dominating set in � :

� �� = = �� ?�?C�� >= �@? � �+� � �

� �� >= �� +� � � � � ? �&� � $According to Theorem 3.1.2 we can express the following additional properties of the (per-fect) dominating set � to the formula implying the membership of the remaining problems toLinEMSOL( � �� ): � � � �

is connected, resp. edgeless, resp. complete, resp. total, resp. cycle–free.

�Note, that in Theorem 3.1.3 the condition that � is fixed is sufficient since in �� no labelslarger than � are allowed implying that we cannot build an MSOL–formula for the solutions.


Observation 3.1.4 For unfixed integer � the problem MINIMUM � –DOMINATING SET (andthus the variants of this problem) is not a problem in LinEMSOL( �� ). Furthermore,MINIMUM � –DOMINATING SET (and the variants of this problem) is not a problem inLinEMSOL( � �� ).

The following result gives a relation of classes of bounded clique width to optimizationproblems in LinEMSOL( �� ):

Theorem 3.1.5 ([40]) Let � be a graph class of clique width at most � such that there is an� �� algorithm constructing a � –expression for every graph � �� . Then, everyLinEMSOL( � �� ) problem on � can be solved in � �� time. A corresponding algo-rithm can be constructed from the logical formula describing the problem and the parsingalgorithm for the class.

So, if the clique width of a class � of graphs is bounded by � and a � –expression can becomputed for all � �� efficiently then every optimization problem in LinEMSOL( �� ) canbe solved on � efficiently. This result gives a unified approach solving many graph theoreticproblems on classes of bounded clique width.

For domination it is an interesting open question if there is an extension � ofLinEMSOL( � �� ) such that � –Domination for unfixed � and � –Domination can be expressedin � without destroying the property of Theorem 3.1.5.

3.1.3 Graph classes defined by two forbidden four–vertex graphs

In [20] we investigate the structure and clique width of graph classes defined by two for-bidden four–vertex graphs. For nearly all combinations we determine whether the cliquewidth is bounded or not; in Figure 3.2 our results are listed. By Theorem 3.1.1 it remains toinvestigate the status of clique width for the following two classes:

1. �� < � �-free graphs,

2. �� co-diamond)-free graphs.

To give the reader a feeling how to prove that a graph class has bounded or unbounded cliquewidth we give one example for both cases.

First, we want to consider a graph class of bounded clique width.

Theorem 3.1.6 ([20]) If � is a prime ( � � ,co-paw)-free graph then � has at most nine ver-tices.

Proof. Assume first that � contains a � � $ ��%!�� .We call a vertex � � �*' � $ ��%!�� an 2 –vertex, 29� � � � 4 � <�� /� , if � �� " � $ ��%!�� 2 .

Let � (resp. �� , �� ) denote the set of 1-vertices adjacent to $ (resp. % , ), let � � � (resp. � � � ,�� ) denote the set of 2-vertices adjacent to $ ��% (resp. $ �� , %!�� ) and let�

denote the set of0-vertices with respect to $ ��%!�� .2

2Note: Since � is �� –free � contains no 3–vertices.

16C

HA

PTE

R3.

SPE

CIA

LG

RA

PHC

LA

SSE

S

� �

� �

� �

� �

� �

� �

� �

� �

� �

� �

�� co-diamond �� co-paw � �

� �

� �

� �

� �

� �

� �

� �

� �

� �

� �

claw

co-claw � � paw diamond

� �

� �

� �

� � � � �� co-diamond �� co-paw claw � � co-claw � � paw diamond � �

�� bcw � bcw

co-diamond � � � � � � � � bcw bcw bcw �

�� bcw bcw �

co-paw � � � � � � � bcw bcw bcw bcw bcw

claw � � � � � � � bcw � bcw � �

� � � � � � � � � � � � � � � � � � � � � � � �

co-claw � � � bcw bcw � � � � � � �

� � � bcw � bcw � � � � � � � �

paw bcw bcw bcw bcw bcw � � � � � � �

diamond � bcw bcw bcw � � � � � � � �

� � bcw � � bcw � � � � � � � �

Figure 3.2: Clique width of �� -free graphs. ’bcw’ (resp. ’ � ’) means bounded (resp. unbounded) clique width. For the four casesmarked with ’?’ it is unknown if the clique width is bounded.


Note, that since � is � � -free, � � � , � � � and �� are stable sets. Moreover, since � isco-paw-free, every vertex �� $ ��%!�� has distance at most 2 to $ ��%!�� .Claim 1.

�and � � � , � � � and �� are homogeneous sets and thus have at most one vertex.

We first show that�

is a homogeneous set. Assume not, and let = �@?N� � distinguished by�+ � � . If =H?C� � then, since �+ � � $ ��%!�� and � is � � -free, � is nonadjacent to one of $ ��%!�� ,say but then =H? � induce a co-paw. If =H? � � then, since � has a neighbor in

� $ ��%!�� , say$ , the vertices = �� $ �@? induce a co-paw — a contradiction in both cases.

Now assume that � � � is no homogeneous set. Let = �@?C�N � � � distinguished by �+ �N � � � .If � � � then � is adjacent to

� $ ��%!�� . Since � �D � � � , � is adjacent to but now =�� ?induce a co-paw. If �� then ? % � induce a co-paw — a contradiction in both cases. Theproof is similar for � � � and �� .Claim 2. � , �� and �� are connected by a join.

Let = �N � and ? � �� . Since ? % = is no co-paw, =H?C� � . The other proofs are similar.

Claim 3.�

is adjacent to every vertex in ��$ ��%!�� .If not, � will contain a co-paw.

Claim 4. � and � � � , = , ? � � $ ��%!�� , = ? , are connected by a join.

Let �3�N � and� �N � � � . Say � is the remaining vertex in

� $ ��%!�� , i.e.� = �@? ��/� � $ ��%!�� .

Since �� ? � is no co-paw, �

� � � .

Claim 5. For all = � � $ ��%!�� holds � � � � 4 or � � � < and � is an edge.

Say = $ . Since � is �� !� 1 � � -free � induces a� � � ( � �� for suitable

�, � � � � .

Assume � � � � < . Since � is no homogeneous set and by Claim 2, 3 and 4 two verticesof � can only be distinguished by vertices from � � � we conclude � � � �

� � for some� . Assume � , � �D � , � � � � , can be distinguished by � . Then, � � � is a co-paw — acontradiction. Thus, � must be an edge.

Altogether � can contain at most nine vertices.

Now, assume that � is �� -free. We consider two cases:

Case 1. � is 1 � -free. Then � � � � ��( � � for some

�, � � � � and thus � � �� < .

Case 2. � contains a 1 � ��$ ��%!�� . Since � is co-paw-free every vertex is adjacent to somevertices of the 1 � . It is easy to see that � , �� , �� , � � � are pairwise connected by ajoin implying � � �� .

�Next, we want to consider a graph class with unbounded clique width:

Theorem 3.1.7 ([20]) ( � � ,diamond, � � ,claw)–free graphs are not of bounded clique width.

The proof of Theorem 3.1.7 is similar to the proofs given in [105]. We have to show that( � � ,diamond, � � ,claw)–free graphs contain graphs with ’good’ grid structure that have un-bounded clique width.


First, we need some definitions: Let � � � �� be a graph and � �� be a

partition of � into red and blue vertices. As follows we define an equivalence relation �� on the vertices contained in �� :

�� if and only if there is no blue vertex which distinguishes

�and � .

The 2-color-width of � , denoted as < �� 6�� , is defined as the smallest number

� � � ,such that there is a partition � �� of the vertices of � into two sets such that� � � � � � �� < � � � � and �� has exactly

�equivalence classes. The following theorem

gives the relation between 2-color-width and clique width:

Theorem 3.1.8 ([105]) For every graph � , if � � �� then < �� 6�� .By this Theorem graphs with unbounded 2-color-width have unbounded clique width, too.

Now, we define a set of special ( � � ,diamond, � � ,claw)–free graphs which have a ’good’grid structure. Let � � be a � � � grid, � � � . For every vertex � occurring in � � at column2 and row � we write �� 2 and �� . We construct a graph � � from � � by thefollowing steps:

1. Replace every edge of � � by a simple path of length three, introducing two new ver-tices which are the internal vertices of the path. Let �3�� denote the resulting graph.

2. Let � be a vertex of degree three in �3�� . Then, there are exactly two neighbors�

, � of� which are in the same row or column in �3�� . Add the edge

� � . This step is repeatedfor all vertices of degree three in � �� . Let � � �� denote the resulting graph.

3. Let � be a vertex of degree four in �3� �� , and let� ��$�$�$ � � � be the four neighbors of � in

� � �� in clockwise order starting from west (i.e.� � , � � (resp.

� ,�� ) are in the same row

(resp. column) with � , �� and ��

). Add the edges� � � and

� � � � . This step is repeated for all vertices of degree four in �3� �� . � � is theresulting graph.

As an example in Figure 3.3 the graph � � is shown. Clearly, every graph � � , � � � , is ( � � ,diamond, � � , claw)–free. It remains to show:

Theorem 3.1.9 For every � � � holds < �� 6�� .Proof. First note that � � ��

contains� �

) � � vertices. Suppose that < �� 6�� . Then there exists a partition � �� of � such that

�

��)�

��

4 ��)

�

��

and �� contains less than � � equivalence classes. In the following we suppose the verticesof � � are arranged in � � )*< rows and � � )*< columns. We say a vertex � is a red row-alternating vertex if � � �� and � has a neighbor

� � � �� in the same row. Analogously,we call a vertex � a red column-alternating vertex if �C� �� and � has a neighbor

� � � �� in the same column.


� � � � � � � � � ��

Figure 3.3: The graph �� .

Case 1. For every 2�� 54 � � � � ��$�$�$ � � � )+<5� there exists a red row-alternating vertex � � inrow 2 .Then, it is easy to see that � � � � �!� ��$�$�$ � ��

are � vertices which belong to � differ-ent equivalence classes of � �� . Therefore this case cannot occur.

Case 2. For some 2 � �54 � � � � ��$�$�$� � � ) <5� row 2 contains no red row-alternating vertex.

Case 2a. All vertices in row 2 are red. We construct a set � of red vertices accordingto the following algorithm:

Algorithm CONSTRUCT �

(1) � � 4; � � � ;

(2) repeat(3) if not all vertices in column � are red(4) then let � be any red column-alternating vertex in column � ;

� � � � � �� ;(5) else

�all vertices in column � are red �

(6) check whether all vertices in columns � . 4 , � . < are red;(7) if ’no’ then let � be any red vertex occurring in column � . �

which has a blue neighbor in column � . � . 4 , � � � � � 4 � ;� � � � � �� ;

(8) � � � . � ;(9) until � � � ) < ;


graph class clique width where

� �� ) � � graphs, � � � �� [105]

� �� ) � � graphs, � � � �� [40]

� �� ) 4 � graphs, � � � unbounded [105]

�� graphs, � � � ��graphs unbounded [105]

bipartite permutation unbounded [22]

cographs� < folk

distance–hereditary� � [68]

partial � –trees� < % 7 � . 4 [43]

permutation unbounded [68]

split unbounded [105]

square grids unbounded [68]

trees� � [43]

unit interval unbounded [68]

Table 3.1: Some graph classes with bounded an unbounded clique width.

Since by construction all vertices of � occur in different equivalence classes of�� we conclude � � � � � � . Therefore, there are at least < � � columns � in�54 � � � � ��$�$�$ � � � ) <5� such that all vertices at columns � and (if � � � ) < ) � . 4and � . < are red. Counting the number of red vertices in these columns and thered vertices in row 2 , we get

� �� /; <�� ) � � . � < � ) < � � ) � < � ) < � . � � � ) < �

4 ��)

�

��

and conclude � �� /; 4 � � � ) � � � , therefore this case cannot occur.

Case 2b. All vertices in row 2 are blue. This case is handled similar to Case 2a above.Now, we reach a contradiction.

�3.1.4 Overview

In Table 3.1 we collect a list of graph classes of bounded and unbounded clique width.See [11] for a survey on the clique width of graph classes defined by three forbidden 1�extensions.

3.2 Graph classes defined by small forbidden subgraphs

Many hereditary graph classes can be characterized by forbidden induced subgraphs or, ifnot, at least a list of small forbidden subgraphs is known (see [23] for an overview of such

3.2. GRAPH CLASSES DEFINED BY SMALL FORBIDDEN SUBGRAPHS 21

classes). In this section we consider � –free graphs where � is a set of graphs containing atmost four vertices. For all of these classes we investigate if MINIMUM DOMINATING SET is��

–complete or if it can be solved efficiently. For the case that only one graph is forbidden(i.e.

�–free graphs) we give the complexity status for arbitrary graphs

�(i.e. not only for

the case � � � � � � �� ).

3.2.1 � –free graphs

At first we consider graph classes which can be characterized by one forbidden inducedsubgraph.

Theorem 3.2.1 Let�

be a graph. If� 1 %��-( � �� for some � � �54 � <�� , � � � � then

MINIMUM DOMINATING SET can be solved in � � � � 7 � � . � � �time on

�–free graphs.

Otherwise, the problem is��

–complete.

Proof. First let� 1#% �6( � �� for some �� 54 � <�� , � � � � and � � � ��

be anarbitrary

�–free graph. We have to show that in this case MINIMUM DOMINATING SET can

be solved efficiently. If � is 1 � -free then a minimum dominating set can be computed inlinear time ([45]). Thus, let � contain 1�� $ ��%!�� . Consider the following algorithm:

Algorithm DOMP4

(1) � � � $ ��%!�� ;(2) while � is not a dominating set in � do(3) begin(4) Let = be a vertex in �(' �� ;(5) � � � � � = � ;(6) end;

By construction ��' � $ ��%!�� is stable and every vertex is nonadjacent to any vertex of the1 � ��$ ��%!�� . Since � is�

–free the while-loop starting in line (2) passes through at most� ) 4 times implying � � � � � . � . Therefore �� . � and a minimum dominating setcan be computed in � � � � 7 � � . � � �

time.

Now, let� 1 % � ( � �� for any �� 54 � <�� , � � � � and � � � ��

be an arbitrary�–free graph. We consider three cases:

Case 1.�

contains a �� .Then bipartite graphs (=

� �� $�$�$ � –free graphs) are�

-free implying the��

–completeness of MINIMUM DOMINATING SET on

�–free graphs ([49]).

Case 2.�

contains a � � or � � or < � .

Then split graphs (= � � �!�� < � �–free graphs) are

�-free implying the

��–

completeness of MINIMUM DOMINATING SET on�

–free graphs ([45]).


Case 3.�

contains no �� , � � , � � and < � .

Since every cycle � � , � � � , contains a < � ��is a forest. Further,

�contains at most

one nontrivial connected component, otherwise�

contains a < � . On assumption this

component is not a 1 � , � � �54 � <�� . Hence,�

must contain a claw implying thatclaw–free graphs are

�–free. Thus, MINIMUM DOMINATING SET is

��–complete

for�

–free graphs ([74]).

�Theorem 3.2.2 Let

�be a graph. If

� 1 % �"( � �� for some �A� �54 � <�� , � � � �then MINIMUM CONNECTED DOMINATING SET can be solved in � � �

� 7 � � � . � � �time on�

–free graphs. Otherwise, the problem is��

–complete.

Proof. The proof is similar to the proof of Theorem 3.2.1. For the case� 1 %��-( � �� for

some � � �54 � <�� , � � � � and � � � �� be an arbitrary

�–free graph we only have to

exchange Algorithm DOMP4 as follows:

Algorithm DOMP4CONN

(1) � � � $ ��%!�� ;(2) while � is not a dominating set in � do(3) begin(4) Let ? be a vertex in �A' �� such that � � ? � � �� < ;(5) Let = � � � �

with =H?C� � ;(6) � � � � � = �@?H� ;(7) end;

Let � be the set of all ? -vertices taken in (4). By construction � is stable and every vertexof � is nonadjacent to any vertex of

� $ ��%!�� . Since � is�

–free the while-loop startingin line (2) passes through at most

� ) 4 times implying � � � � � . < � � ) 4 � < � . < . Since� is connected � � �� < � . < and a minimum connected dominating set can be computedin � � �

� 7 � � � . � � �time.

The��

-completeness proof for the case that� 1 % � ( � �� for any � � �54 � <�� ,� � � � is analogously to the proof of Theorem 3.2.1 using [67, 97, 111, 124].

�

3.2.2�

� !�� –free graphs

Next, we investigate graph classes which can be characterized by exactly two forbiddensubgraphs with at most four vertices. In Figure 3.4, 3.5 and 3.6 all graphs with at least twoand at most four vertices are shown.

Immediately, by Theorem 3.2.1 we conclude

Observation 3.2.3 The following propositions are true:


� � � �� < ��

Figure 3.4: Graphs containing exactly two nodes.

�

� �

�

� �

�

� �

�

� �� co- 1 � � � 1 � ��

Figure 3.5: Graphs containing exactly three nodes.

� �

� �

� �

� �

� �

� �

� �

� �

� �

� �� co-diamond �� < � � �� co-paw

� �� 1 ��

� �

� �

� �

� �

� �

� �

� �

� �

� �� claw � �� co-claw � � � � �

� �� paw � �� diamond

��

��

��

��

��

��

� �

� �

� � � � �

��

��

��

Figure 3.6: Graphs containing exactly four nodes.


1. If 2*� �54 � <�� 4 � � 4 4 � MINIMUM DOMINATING SET (MINIMUM CON-NECTED DOMINATING SET) can be solved in polynomial time on � � -free graphs.

2. If 2C� �� 4 <�� 4 � � 4 � � 4 � � 4 � � 4 � � MINIMUM DOMINATING SET (MINIMUM CON-

NECTED DOMINATING SET) is��

-complete on � � -free graphs.

By this observation we need only to investigate �� -free graphs for 2 , � �� 4 <�� 4 � � 4 � � 4 � � 4 � � 4 � � , 2 � .

Immediately we get some��

-complete cases by

Observation 3.2.4 The following propositions are true:

1. �� -free = � � -free for all 2 � �54 ��$�$�$ � 4 � � .2. �� -free = �� -free for 2 4 � � 4 � � 4 � � 4 � .

3. If � 2 � � � � � � 4 � � 4 �� 4 � � 4 � � � � 4 � � 4 � � � � 4 � � 4 � � � � 4 � � 4 � � � � 4 �� 4 � � � then �� -free �� -free.

4. Split graphs are �� -free.

Theorem 3.2.5 ([132]) For every �N� � MINIMUM DOMINATING SET is��

-complete on� � �!��$�$�$ �� % 7 � –free planar bipartite graphs with maximum degree three.

Corollary 3.2.6 MINIMUM DOMINATING SET is��

-complete on �� !�� –free =�� !� co-claw � paw � diamond � � � � -free graphs.

By the results from subsection 3.1.3 we immediately get some cases where the clique widthis bounded and thus MINIMUM DOMINATING SET can be solved in linear time:

Theorem 3.2.7 ([20]) For the following parameters � 2 � � � �� –free graphs havebounded clique width and a � –expression can be computed in linear time: (6,9), (6,12),(9,15), (9,16), (12,13), (12,15).

Corollary 3.2.8 MINIMUM DOMINATING SET can be solved on ( < � , co–claw)–free

graphs in � � � � � time.

Proof. Let � be a ( < � , co–claw)–free graph. If � is �� –free then by Theorem 3.2.7

MINIMUM DOMINATING SET can be solved in linear time. If � contains a � � this � �dominates the whole graph since � is co–claw free. Thus, �� )� � and a minimumdominating set can be computed in � � � � � time.

�In [27] the authors show that every connected 1 � –free graph has a dominating clique ora dominating 1 � . Using this result they get a polynomial time algorithm for MINIMUM

DOMINATING SET on � < � � � � � –free graphs.

Theorem 3.2.9 ([27]) MINIMUM DOMINATING SET can be solved on � < � � � � � –freegraphs in � � � � � time.


� � �

�%

� � ��

��

��

� �$

� �� '9 �� " �� ' �� A' ��

Figure 3.7: Hanging of the edge �� .

Results on (claw,�

)-free graphs

By Theorem 3.2.7 MINIMUM DOMINATING SET can be solved on (claw, � � )–free and (claw,co-claw)–free graphs in linear time since these classes have bounded clique width and a � –expression can be computed in linear time. For (claw, � � )–free one can see this also directly:

Observation 3.2.10 Every (claw, � � )–free graph is a disjoint union of induced paths andinduced cycles greater than three.

Next, we want consider (claw, < � )-free graphs.

Theorem 3.2.11 For (claw, < � )-free graphs MINIMUM DOMINATING SET can be solved

in linear time.

Proof. Let � � � �� be a (claw, < �

)-free graph. If � is 1 � -free then a minimum dom-inating set can be computed in linear time ([45]). Therefore, let � contain a 1 � ��$ ��%!�� .We consider the hanging of the edge �� (see Figure 3.7). The defined sets have the followingproperties:

(P1) � is stable since � is < � -free.

(P2) � and�

are complete since � is claw-free.

(P3) � �>= � " � � � 4 for all = � � � � � �since (P1) and � is claw-free.

(P4) % � � , $ � � .

We consider two cases:

Case 1. � � � �+< .Let � � ��$�$�$ � � � � . We partition � � � � �

according to their neighborhood in � :

� � � � � � � � $�$�$ � � � � � � 7 �� 2 4 ��$�$�$ � � � � � � 7 � ��( � $


Since � is < � -free and � is stable we conclude � �� ! � � for all 2 , � � �54 ��$�$�$ � � � ,

2 � . Now, we distinguish two subcases:

Case 1a.� �� .

Let �� $�$�$ � � � � � � � � $�$�$ � � � and set � � �� $�$�$ � � �@� . Clearly, � is

a dominating set in � . Suppose, that there is a dominating set �� in � with� � � � �

. Since � is stable and � �>= �#" � � 4for all = � ��

by (P3) weneed at least

�vertices to dominate the vertices in � , i.e.

� �� " � � � � � � � � $But now �� ( � � � leads to a contradiction implying �� . 4 .

Case 1b. There are 2 , � � �54 ��$�$�$ � � � , 2 � such that � � " � � and � � " � � � � � � .Let �� $�$�$ � � � � �� $�$�$ � � � with � �� and ��

and set � � � � � ��$�$�$ � � �@� . By construction � will dominate all vertices in

� ��

�� (' � � 7 � $

Let = � � � 7 � . If = � � then = � � � � since � is complete by (P2). If =� � � then= � � � �and = � � � � since otherwise

� = �� will induce a < � . This

proves that � is a dominating set in � . Analogously to Case 1a one can showthat � is minimal, i.e. ��

.

Case 1c. � " � � �� .This case cannot occur since % � � has the neighbor $ in � .

Case 2. � � � 4.

In this case � � � %!�� is a dominating set in � implying �� < .This computation of a minimum dominating set can clearly be implemented in linear time.

�Note, according to our proof of Theorem 3.2.11 one can easily develop a robust algorithm tosolve MINIMUM DOMINATING SET on (claw, < �

)-free graphs.

For investigating negative results on the complexity of domination problems on sub-classes of claw-free graphs it is useful to consider line graphs — a proper subclass of claw-free graphs. More information (in particular a characterization by forbidden induced sub-graphs) one can find in the survey [23].

For a graph � � � �� the line graph � ��

has as vertices the edge set � of � and twovertices �� , � � � of � ��

are adjacent if they have a common endpoint, i.e. � � " � � .Clearly, every line graph is claw-free because if �� is a 1 � in � ��

then �� and � �must contain different endpoints of � .

We call a set �� an edge dominating set in � iff �3� is a dominating set in � �� ,

i.e. for all � � � either � � �3� or � " � � for some � � �3� . Let MINIMUM EDGE DOMI-NATING SET be the problem of computing an edge dominating set of minimum cardinality.


graph class time complexity where

bipartite permutation � � � � . � � [118]

bipartite graphs with maximum degree 3��

–c [128]

block graphs LIN [71]

claw-free chordal graphs � � � �

[72]

co-chordal � � � � �� . � � [118]

locally connected claw-free graphs � � � � � [72]

perfect claw-free graphs��

–c [72]

planar bipartite graphs��

–c [72]

planar cubic graphs��

–c [72]

planar graphs with maximum degree 3��

–c [128]

trees LIN [104, 128]

� -trees�

[37]

Table 3.2: Complexity of the Edge Domination Problem on special graph classes. Abbrevi-ations:

�for polynomial time, LIN for linear time and

��–c for

��–complete.

This problem has interesting applications in telephone switching networks (see [128]) andamong other things the complexity is known for the special graph classes listed in Table 3.2.We show

Theorem 3.2.12 MINIMUM DOMINATING SET is NP-complete on (claw,diamond, �� -free graphs.

Proof. This result is a consequence of the��

-completeness of MINIMUM EDGE DOMI-NATING SET on � � -free bipartite graphs with maximum degree three (see [128]3). For thiswe need to investigate what � ��

is�

-free,� � � � �!� � �!� diamond � , means in � .

Observation 3.2.13 The following properties are true:

1. 0 �� if and only if � �� is � � -free.

2. � is � paw � diamond � � � � -free if and only if � �� is diamond-free.

3. � is � � � � diamond � � � � -free if and only if � �� is � � -free.

Proof.

1. Let �� be a � � in � �� . Then �� cannot induce a triangle in � since

�� is adjacent to all elements in� � � �� . Therefore, � � �� contain a common

3In [128] it is only written that MINIMUM EDGE DOMINATING SET is��

-complete on bipartite graphswith maximum degree three but the graph used in the reduction proof is � � -free, too.


��/��

claw co-claw �� paw diamond� ��

–c LIN LIN��

–c��

–c��

–c��

–c��

–c��LIN

��–c LIN � � � � � ��

–c LIN LIN � � � � �claw LIN LIN

��–c LIN

��–c LIN

��–c

��–c

co-claw��

–c � � � � � LIN��

–c��

–c��

–c��

–c��

–c

�� –c

��–c

��–c

��–c

��–c

��–c

��–c

��–c

paw��

–c LIN LIN��

–c��

–c��

–c��

–c��

–c

diamond��

–c LIN��

–c��

–c��

–c��

–c��

–c��

–c� ��–c � � � � � ��

–c��

–c��

–c��

–c��

–c��

–c

Figure 3.8: MINIMUM DOMINATING SET on �� -free graphs. Abbreviations: LIN forlinear time and

��–c for

��–complete.

vertex = �� " � " � � . Since �� is adjacent to � �� , vertex = also is an endpointof �� implying � �-, . �>= � � � . The other direction is clear.

2. Let �� be a diamond in � �� with nonedge between � and �� . Since

�� is a clique in � �� these edges are either a claw (note: not necessary an

induced claw!) or a triangle in � . In both cases the remaining edge � � will give a paw.Thus, � contains a paw, diamond or � � .

3. Let �� !�� be a � � in � �� . Then, �� must also form a 4-cycle in

� . A 4-cycle induces in � a � � , diamond or � � .

�By this observation line graphs of � � -free bipartite graphs with maximum degree three are(claw,diamond, � � � � � � -free which settles the proof of Theorem 3.2.12.

�3.2.3 Overview

In this subsection we want to summarize the obtained results.

For one forbidden subgraph Theorem 3.2.1 gives a complete answer to the complexitystatus of the problem MINIMUM DOMINATING SET.

For two forbidden subgraphs with at most four vertices the complexity status is eitherpolynomial or the status is given in Figure 3.8.

Altogether, we are able now to give the complexity status for all � -free graphs where �is a set of graphs containing at most four vertices.

Corollary 3.2.14 Let � be a set of graphs containing at most four vertices. Then, eitherMINIMUM DOMINATING SET can be solved in polynomial time on � -free graphs or �is a superclass of one of the following graph classes and MINIMUM DOMINATING SET is��

-complete:

3.3. THE CLASS �� !�� — CONTRACTION OF HOM. SETS 29

� � � �!� < � �-free graphs,

� (claw, � � , diamond, � � )-free graphs,

� ( � � , � � )-free graphs.

3.3 The class� �

�� 4� — contraction of homogeneous

sets

In this section we consider the relation of homogeneous sets to the minimum � –dominatingset problem for graphs without an underlying tree structure. We show that one can reduce ahomogeneous set to one vertex or to a so–called meta–vertex consisting of two nonadjacentvertices with � –value one. This reduction together with modular decomposition leads toan � �� time algorithm for computing a minimum � –dominating set for graphs whichcan be generated from the one–vertex graph by a finite number of homogeneous extensions(substitution of an arbitrary graph into a vertex) and by attaching pendant vertices (leaves).For distance–hereditary graphs — a proper subclass of this graph class — we even get alinear time algorithm.

In [92] the authors reduce the total dominating set problem to the dominating set problem.They show that for any graph class which is closed under adding false twins an � �� algorithm for the dominating set problem leads to an � ��

algorithm for the totaldominating set problem (see Theorem 2.1.2). Therefore we get a linear time algorithm forthe total dominating set problem on distance–hereditary graphs.

As mentioned in Section 3.1 the existence of a linear time algorithm for the minimumdominating set problem on distance–hereditary graphs can be proved using another ap-proach: In [68] the authors consider the clique width of some perfect graph classes. Theyshow that for distance–hereditary graphs the clique width is bounded by three and that a 3–expression can be computed in linear time. As consequence (see Theorem 3.1.3 and 3.1.5)MINIMUM DOMINATING SET can be solved on this class in linear time.

This section is organized as follows. After presenting basic definitions and notationswe will outline the idea of the algorithm presented in [108] for homogeneous extensions oftrees. We hope this gives a better understanding of the more general reduction presented insubsection three. After that this theorem is used to develop efficient algorithms for solvingthe minimum � –dominating set problem on �� and distance–hereditary graphs.Finally, some open problems for further research are listed.

3.3.1 Homogeneous Extensions of Graphs

Recall, a set� � � is called homogeneous iff any pair of vertices of

�has the same

neighborhood outside�

:

� � � " � �(' � � �� " � �A' � �for all

� � � � � $


A homogeneous set�

is proper (or nontrivial) iff� � � � � � � � � . With � ��

we denotethe set of all maximal proper homogeneous sets of � .

Let�

be a proper homogeneous set of � containing at least two vertices and let � � � �.

Then the graph �� obtained from � by deleting

� ' � � �9� , i.e. contracting�

to a representing vertex � � , will be called the homogeneous reduction of � (via�

). Thefollowing is easy to prove.

Observation 3.3.1 Let � � � �� be a graph,

�a proper homogeneous set in � and

� � � �. Then the distances in � and ��

fulfill the following properties:

� 4 � � . �>= �@? � � . � �>= �@? � for all = , ?C� �(' �,

� < � � . �>= � � � � . � �>= � � � �for all = � �(' �

,

� � � � . � � � � � � � < for all� � , � � �

if � is connected.

Conversely, for a connected graph � of at least two vertices the homogeneous extension��

of � via a graph�

in � is the graph obtained by substituting � by�

suchthat the vertices of

�have the same neighbors outside

�as � had in � . Note that it is

necessary for structural reasons to have at least two vertices in � since otherwise each graphcan be represented as the homogeneous extension of a single vertex by itself.

Recall, a graph � � � �� is called prime graph iff � , � and the singletons of � are the

only homogeneous sets in � . For a connected graph � we define �� as:

� � if � has at most two vertices.

� If � �� is a partition of � then ��

is the graph obtained by contracting eachproper homogeneous set of � ��

to a representing vertex.

� Otherwise we define �� .

Two homogeneous sets� � , ��

overlap iff their intersection and their mutual differencesare nonempty. A homogeneous set

�is overlap–free iff there is no other homogeneous

set overlapping�

. Since for any overlap–free homogeneous set� � there is exactly

one minimal overlap–free homogeneous set� � containing

�properly we obtain a parent

function by � � ��-F �� . Thus, using � as root, this gives a tree of homogeneous setswhich is called the module tree ��

. This tree can be computed in linear sequential time(cf. [106], [34]).

In [108] we consider the relationship between the maximal proper and maximal overlap–free homogeneous sets.

Lemma 3.3.2 ([108]) Let � be a graph with at least two vertices.

� If � �� is a partition of � then � ��

is exactly the set of maximal proper overlap–freehomogeneous sets of � .


��

��

�1

2

3 56

4

�

��

1

5

4

�� 54 � <�� /� � 4 �

��

��

1

2

3 5

4

�� 4 � � ��

��

� �� 54 � <�� /� � � � � � � � � �/� � � � � � ��7

8

� � ��

��

��

� � ��

�6��7

8

� � ��

��

��

� � ��

�7

8

��9

10

��

��

Figure 3.9: Example for the operations �� , �� and �� .

� If � �� is not a partition of � then the maximal proper overlap–free homogeneous

sets of � are exactly the connected components � ��$�$�$ � � � of the complement � , and� �� .

Thus, using modular decomposition, we can compute in linear time homogeneous sets� � ��$�$�$ � � � such that� � ��$�$�$ � � � � is a partition of � and such that reducing each

� � to arepresenting vertex leads to ��

(if � �� is a partition of � ��

then we take � �� ,

otherwise � � , � � �� is a partition of � �� and ��

).

Next we want to describe graph classes which can be generated from some basic graphsby a finite number of certain operations. Let � be a set of graphs and � be a set of operationsdefined on graphs. Then ��

is defined as:

� 4 � Every graph of � belongs to �� .

� < � If � � is an � –ary operation from � and �� $�$�$ ��% are graphs from �� then

� � �� $�$�$ ��% � belongs to �� , too.

Many known graph classes can be described in this way. Indeed, any graph class whichcan be characterized via dismantling schemes can be represented by the above –notation.Thus, trees, chordal graphs, strongly chordal graphs, dually chordal graphs, distance–here-ditary graphs, HHD–free graphs can be described by the –notation. For instance, dist-ance–hereditary graphs are exactly the graphs of �� !�� (cf. [24, 77]). Since anypair of twins is a homogeneous set distance–hereditary graphs are a (proper) subclass of �� .

In Figure 3.11 the inclusion hierarchy of some graph classes is shown. For definitionsand characterizations of these graphs we refer to [23] for an overview, to [12] for duallychordal graphs, to [13] for homogeneously orderable graphs and to [54] for homogeneousgraphs. Here we only want to define the � � –operation. A vertex

� �� 4 � is a maximum


�1

�1

�2

� �1

�2

� �3

� �1

�2

�3

�4

��

��1

��3

�4

��

5

6

��1

��3

��

5

6 �

��

8

7

��1

��3

��

5

6 �

�� 9

8

7

�910

��1

��3

��

6 �

��

8

7

�910

5

�11

Figure 3.10: Generation of a graph in �� which is not in � �� trees�.

neighbor of a vertex � iff � � � � 4 � � �� < � , i.e. � � � � 4 � � � � � � 4 � for all �!� � �� 4 � . If� has a maximum neighbor then it is called extremal. Now by � � we denote the operationof adding an extremal vertex.

By the recursive definition of �� there is a corresponding generation sequence to

each graph � of �� : We call � � � �� $�$�$ � � � �� a generation sequence of� iff

� 4 � � � is the one vertex graph with (initial) vertex �� ,� < � � � � �� if < � � �� , or � � is obtained by attaching = as pendant vertex

to � � where �� = � , 2 � �54 ��$�$�$ � � � ,� � � � � � .

Note that by the definition of homogeneous extensions � � is always obtained by attaching apendant vertex to the vertex of � � .The reverse order of a generation sequence is called reduction sequence.

Observation 3.3.3 ([36]) For homogeneous sets� � and

��of a graph � the following prop-

erties are fulfilled:

1.� � " ��

is homogeneous.

2. If�� then

� � ' ��is homogeneous.

3. If� � " �� then

� � � ��is homogeneous.

Lemma 3.3.4 ([109]) Let � be a graph in �� .� 4 � If � is a pendant vertex in � then �*) �C� �� !�� .


homogeneously orderable graphs ��

�� dually chordal�

dually chordal �� dhg

�� !�� !� trees�

trees ��

homogeneous graphs

Figure 3.11: Inclusion hierarchy of some graph classes.

� < � If�

is a proper homogeneous set in � and � � � �then ��

�� .Proof. Let �

� � � �� $�$�$ � � � �� be a generation sequence of � .

� 4 � First assume that � is the initial vertex, i.e. � � � � �� . Recall that �� is obtainedby attaching = as pendant vertex to � , � � � = � . Since � is a pendant vertex in � nofurther operations are performed on � . Thus, by starting with = as initial vertex, we canrearrange � to

� � � � � � � � � ��$�$�$ � � � �� = � � $Now let 2 be the position in � such that �C� � �� but �� .If � is attached as pendant vertex in step 2 , i.e. � � � �� then

� � � � � � � � � � ��$�$�$ � � �� 87 �� 87 � � ��$�$�$ � � � �� is a generation sequence of � , too. Thus, �*) � is in �� .If � �� and � �� D< then � is an isolated vertex in � � �� and . � � �� = � . Thuswe can generate � by

� � � � � � � � � � ��$�$�$ � � �� ' � �� = � � � ��87 � � � �87 � � ��$�$�$ � � � �� i.e. � can be attached as pendant vertex. The assertion follows by the preceding argu-ments.

� < � Assume for the contrary that � is a vertex minimal graph in �� !�� such thatthere exists a proper homogeneous set

�in � with �� .

Then clearly � � � �+< and� � � .


At first we consider the case � � � � �+< . Supposing � � " � � we get a contradiction tothe minimality of � by

�� $Therefore

� " � � � .If

� �� we have nothing to show since

�� $So let

� � � � '�� . Analogously we may assume� � � � ' � � since

otherwise we get

�� @� � � � � $Note that by Observation 3.3.3 each of the sets

� � , ��and

�(� � � is homogeneous. Butnow

��

gives a contradiction.

In the case � � � � 4we conclude �/�9� �

by (1) since � is a vertex minimal counterex-ample. But then

�� *) � �� ' � � � � � � � � �which settles the proof.

�A direct consequence of the last lemma is

Observation 3.3.5 ([109]) Let � � � be a graph in �� !�� . Then ��

hasat least one pendant vertex.

Furthermore, we obtain a simple recognition algorithm.

Corollary 3.3.6 ([109]) Graphs of �� can be recognized in � �� time.

Proof. The following procedure recognizes graphs of �� :� 4 � Compute the module tree � ��

and let � be its root.

� < � If � � � � � �� then

� Homogeneously reduce each�

of � to a representing vertex � � establishing��

.� If ��

has no pendant vertex then stop — � is not in �� .


� Otherwise, as long as possible dismantle all pendant vertices. Let � � be the ob-tained graph.

� If �� then stop — � is in �� . Otherwise go to step � 4 � with� �� .

� � � Otherwise ( � �� is not a partition of � ) stop — � is in �� .

Since the computation of the module tree takes linear time the whole algorithm runs in� �� time.

�As already mentioned in the introduction the minimum dominating set problem on distance–hereditary graphs can be solved in linear time since distance–hereditary graphs have boundedclique width (see [68]). For graphs in �� this approach cannot be used:

Observation 3.3.7 ([109]) The class �� is not of bounded clique width.

Proof. Let � � � �� be a graph of clique width �� < and = � � . With � � ! = we

denote the graph � � � � = � �� = �� N� � � � (see Chapter 2). Then it is easy to see that��6!N= � �� and that � �6! = has clique width � , too.

�3.3.2 The � –domination set problem on homogeneous extensions of

trees

In [108] we have seen that homogeneous sets behave very well with respect to dominationproblems. For a better understanding of the reduction presented in the next subsection wewill at first outline the idea of the algorithm presented in [108] for homogeneous extensionsof trees.

Let � � � �� be a graph with radius function � � � � �

. For every homogeneous set�of � we denote by

� � the set of vertices of�

with � –value zero. By definition� � must

be included in any � –dominating set of � . Thus, if� � � we may reduce

� � to a singlevertex (which is adjacent to all neighbors of

� � ), and at the end of the algorithm we replacethis single vertex by the whole set

� � . So we may assume � � � � � 4. But now each proper

homogeneous set�

of � can be � –dominated by at most two vertices (recall � � �� !< � .Hence we transform � into a graph �3� consisting of vertices of the following two types:

� 4 � If there is a vertex � of�

which � –dominates�

in � then�

is � –dominated by a singlevertex. In this case � � is a usual vertex. We define � �� GF � � �� C� � � .

� < � In all other cases � � is a meta–vertex consisting of two nonadjacent inner vertices � ��

�

such that:If

� � � then � �� and � ��

� � 4, otherwise � ��

�

� � 4.

An example of this transformation is given in Figure 3.12. It is easy to see that an optimalsolution for the � –dominating set problem on � can be constructed from an optimal solutionfor the � –dominating set problem on �3� .


� � �K�� ! M K�� ! MK�� ! M�

� �K�� MK� � � � M� ��

K�� MK�� ! MK�� ! MK� � � � M

� �K�� ! MK�� ! M� �

�K�� M�� K�� M

K�� "( M� �K�� MK�� "( M� K�� ! M�

�K O � � � M

� �

� �K �� ! MK �� #" M$

�K&%(' � � M

� �K&% �) � ! MK&% �) � ! M

� �K&% �* � ! MK&% �* � ! M

� �K&% �+ � ! MK&% �+ �"( M �K&%(, � � M

� K&%(- � ! M

�K&%(. �"( M�K&%�/ �#" M

�

�10

Figure 3.12: An example for the reduction of maximal homogeneous sets to vertices ormeta–vertices. �>= �@2 � means: vertex = has � –value 2 . Rectangles indicate homogeneous sets.


��

��

K ! �� M��K � �� M��K ! �� M��

K ! �� M�� K � �� M��K � �� M��

�� K ! �� M��

K � �� M��K ( �� M��

� �K ! �� M��

K � �� M��K ( �� M��

� � � � � �

�� K ! �� M��

K � �� M��K ( �� M��

��

K ! � ! M��K � �� M��

��

�K ! � ! M��

��

��K � �� M��K � �� M��

�K � �� M��

� �

� � � ��

Figure 3.13: An example to the algorithm presented in [9]. � 2 � � � � means: vertex � has� –value 2 and –value � .

So, for homogeneous extensions of trees, we obtained a tree � of vertices and meta–vertices for which the � –dominating set problem can be solved in linear time. The idea ofthe algorithm is similar to the one of [116] and [9] for computing � –dominating sets in trees(the algorithm given in [9] works for dually chordal graphs, a proper superclass of trees). Ineach step the algorithm takes an arbitrary leaf = of the current tree � (i.e. the tree consistingof all unprocessed vertices) and decides if = must be added to the current dominating set

�.

The basic rule is to choose vertices for domination closest to the root, i.e. as long as possiblewe remove leaves without adding these ones to the current dominating set

�. After that, =

is deleted and certain parameters of the father of = will be updated. If the actually graph isempty

�is a minimum � –dominating set for the given graph.

As parameters for the vertices we use two values: � �>= �, �>= �

. Initially � �>= �is the given

� –value and �>= �will be initialized with B . In each step for the parameters of each leaf = of

the current tree � the following properties hold:

� � �>= �represents the distance within = and all still undominated vertices of � � must be

dominated in � ,

� �>= �indicates the minimum distance of = to a member of the current � –dominating set�

in � � .So if �>= � �

� �>= �then = is already � –dominated by

�. If �>= � ; � �>= �

then = is � –dominatedby some vertex � in the current tree if its father ? is � � –dominated by � where �&� � ? � � � �GF � � � ? � � � �>= � ) 4 � . This works well for usual vertices as proved in [9]. An example of thisalgorithm is given in Figure 3.13.

For meta–vertices we cannot use the above technique. Indeed, it is impossible to decideduring the removal of a leaf with a meta–vertex as father whether this leaf has to be added tothe � –dominating set or not. To illustrate this problem consider the example in Figure 3.14.


�

� �

� �

�

�

� �

� �

� �

�� 4 ��

��$ � 4 �

��%!� 4 �

�� 4 �� 4 �

��$ � 4 �

��%!� 4 �� 4 �

Figure 3.14: The problem with meta–vertices.

In the left graph� �� is the unique minimum � –dominating set. On the other hand, in the

right one there is no minimum � –dominating set containing .To avoid this decision we introduce four parameters � �� for meta–vertices �

describing all possibilities for local � –domination. Hereby, � � and � � contain vertices suchthat � � is completely � –dominated by

� � � � resp.� � � � . The difference between these

two sets is that � � must contain at least one (inner) vertex of � whereas � � must not containany vertex of � . On the other hand, the sets � � and � � contain vertices such that ��9' � ��is � –dominated but not � itself. Note that these sets contain only already processed vertices,i.e. vertices from �� .

Here we will use the same technique to obtain a reduction theorem for homogeneoussets which is the base for the polynomial time algorithms of the � –dominating set problemon �� !�� and distance–hereditary graphs presented in the following subsections.Since we have two types of vertices it is convenient to define a more general � –dominatingset problem considered in the next subsection.

3.3.3 The generalized � –domination set problem and homogeneous sets

As mentioned in the previous subsection we define a more general � –dominating set problemwhich additionally reflects the recursion. We are given:

� Finite disjoint sets � — the set of all vertices — and � , a set of markers.

� The set of current vertices � �� 54 � <5� � �� . For each � � � we write � �for �� 4 � , � for �� < � and � for

� � �� .� The current graph � � � ��

such that for all � � � the vertices � � , � are false

twins in � , i.e. �� . We call the elements of � meta–vertices.

� A radius function � � �� B+� .� An initial set � � � ' � .


� For each meta–vertex � we are given sets � � , � � , � � , � �� ' � � � � � �� which fulfill the following properties:

– � � " �� , � � " � � � � � , �� " � � .– � � � � � � � � ��) < , � � � � � � � � ��) 4 , � � � � � � � � ��) 4 . We call the set � 2 –optimal

iff � � � � � � ��) 2 , � � � � �� , 2 � �54 � <5� .– For distinct meta–vertices = , ? the sets � � � � � � � � � � � and � � � � � � � � � � �

are disjoint.

– The sets � � , � � and � � need not to be defined, � � is always defined.

Further — if � � is defined — a value � �� B+� is given. We call � � , � � , � � ,

� � the set parameters of � .

A subset of � is called an (abstract) dominating set iff the following properties arefulfilled:

� 4 � ��* .

� < � (Domination of the meta–vertices)For each meta–vertex � one of its defined set parameters is completely contained in .If neither � � � nor � � � , then the meta–vertex must be 1–dominated in � , i.e. �� " � .

� � � (Domination of the vertices)For each � � � � at least one of the following properties is fulfilled:

(a) � is � –dominated in � , i.e. � . �� E " � � �� .

(b) There exists a meta–vertex = such that

� . �� = � .��

� � " = � �4 � " = � �� * �� >= � � " = � �� * � � � �� $

It is easy to verify that with � � and � � � � we get the usual � –dominatingset problem.

In the following we have given a generalized � –dominating set problem and a (in � )homogeneous set

�with < � � � � � � � � , such that =��

for each meta–vertex = with= " � � . Let� � be the set of vertices of

�with � –radius zero. We now describe the

reduction of�

to a vertex or a meta–vertex.

For each meta–vertex � of � let � �� denote the set of defined set parameters associatedto � , i.e. � �� . Note � �� since � � is always defined. With � � � �we denote the set of all set unions of the parameters of all meta–vertices of

�, i.e.

� � � � � � � � � � � " � � �� " � � $


For a set � in � � � �and a meta–vertex = of

�we denote by � �>= �

the set parameter of =contained in � , i.e. � �>= � �� with � �>= � �� . Let � � � �GF � � � �� . Aset � of � � � �

with � � � � has property

�:1 4 � iff � dominates�

and � " � � ,�:1 < � iff � dominates

�and � " � � ,

�:1 � � iff � " � � .Thus each �:1 4 � –set is a �:1 � � –set, too.

At first we consider the case that�

does not contain vertices with � –radius zero, i.e.� � � .

Case 1.� � � and there is a set � in � � � �

such that � � � � and � dominates�

.

If there is a �:1 4 � –set � in � � � �then let = �� " �

. We add � ' � = � to � , reduce�to the vertex = and define � �>= � � � .

If there are no �:1 4 � –sets but there is a �:1 < � –set � and a �:1 � � –set � � then we reduce�to a meta–vertex

�with the following set parameters: �� , �� *� , �� not

defined and �� = � , where = � �arbitrary.

Finally, if there are no �:1 4 � – and �:1 � � –sets but there is a �:1 < � –set � then we reduce�to a meta–vertex

�with the following set parameters: �� , �� not defined

and �� = � , where = � �arbitrary.

In Figure 3.15 is given an example for Case 1. The instance � � of the � –dominating setproblem can be transformed into an abstract dominating set problem �

with the followingset parameters (the details of the computation of �

are given in section 3.3.4)

� ?�� $ ��%�� ? � �� $ ��%��

� �� n.d. n.d.

�� n.d. n.d.

Using our reduction relative to the homogeneous set� � � � = �@?H� we obtain � � ��

� � � $ ��%!�@? �� as an element of � � � �with minimum cardinality. Since � is a �:1 4 � –set

we reduce�

to the vertex ?�� " �, add � ' � ? � � to � and define � � ?�� . After that

it is not hard to see that the result � � leads to the minimum � –dominating set� $ ��%!�@? ��

in � � .Before considering the remaining cases we show the correctness of Case 1. Note that

the remaining correctness proofs can be performed in a similar way and hence are left to thereader.

Lemma 3.3.8 ([109]) The reductions in Case 1 are correct.


� � � ��

� ��

��

��

��

� � ��

��

� � ��

��

� ��

� ��

K�� "( MK�� ! MK� � ! M

K�� ! MK�� ! M!K&% � ! M

K�� ! M K�� ! MK� � ! M

K�� ! M

K�� ! M K�� ! M

K� � � ! M K� � � ! M� � � �

� �

� �

K�� ! MK�� ! M K&% � ! M K�� ! M

��

�

� �

� � �

�

� � � �K�� ! MK�� ! M K&% � ! M K�� "( M

� � �

� � ��

� ��

� �

��

� � � ��

Figure 3.15: An example for Case 1. � �@2 � means: vertex has � –value 2 .

Proof. At first we have to show that the reduction gives a generalized � –dominating setproblem for the reduced graph �3� . If there is a �:1 4 � –set � then

�is reduced to a vertex =

and hence there is nothing to show. So consider the case that�

is reduced to a meta–vertex�. Obviously, �� is defined, � �� ) 4 and, provided � � is defined, � � � � � �� &) 4 .

Since the sets � , �� are set unions of set parameters of�

these sets are disjoint from anyunion of set parameters of a meta–vertex outside

�. Further, since all set parameters of

meta–vertices contain only vertices outside the current graph (up to the meta–vertex itself)no set parameter of any meta–vertex of � ' �

contains any vertex of�

. Thus in �3� the setparameters of different meta–vertices remain disjoint.

Next we have to show that any minimum � –dominating set of � � is a minimum � –dominating set in � too. First consider the case that there is a �:1 4 � –set � . Then

�is

reduced to a vertex = and � �>= � �in �� . Furthermore, � ' � = � is added to � . Thus any

minimum � –dominating set in �� must contain � and hence, is a � –dominating set in �too. Now suppose that is not minimum in � , i.e. there is a set � � –dominating � such that� �� . By definition we get � � " � � � � . Let �� ' � �� . Obviously � � is a� –dominating set in �� and hence � � � � � � � . But now � � � � �� ) � . � � � � �� gives a contradiction. Thus is a minimum � –dominating set in � .

Now consider the case that there are no �:1 4 � –sets. Here�

is reduced to a meta–vertex�. Let be a � –dominating set of � . By definition, must include at least one of the set

parameters of�

, and if neither �� nor �� is contained in then � � �#" � . Since theset parameters of

�are defined via � , � � the set � –dominates � too. If is minimum in


Algorithm MARK

� �64 � If � � is 2–optimal then we mark � � .� � < � Else, if � � is 1–optimal and = has a neighbor ? in

�which is a meta–vertex and

either

� � � is 1–optimal and � � is not 2–optimal, or� � � is optimal (i.e. neither � � is 2 –optimal, 2 4 � < , nor � � and � � are 1–

optimal),

then we mark � � .� � � � Else, if � � is 1–optimal then we mark � � .� � � � Else, if � � is 1–optimal then we mark � � .� � ��

Else, if � � is 1–optimal then we mark � � .� � �

�Otherwise we mark � � .

�� then is minimum in � which can be proved as before.

�Next we consider the time bound of the reduction in Case 1.

Lemma 3.3.9 ([109]) It can be checked in linear time whether there are �:1 4 � – �:1 � � –sets.Furthermore, such sets can be computed within the same time bound.

Proof. First note that property �:1 � � can be tested in linear time since we have only to checkwhether � � or � � is optimal for a meta–vertex = of

�.

Now we want to check properties �:1 4 � and �:1 < � . For each meta–vertex = we mark a setparameter of = according to algorithm MARK.

Let � be the union of all marked set parameters. Note that � �>= � � � via rule � � < �immediately implies that = is dominated by � .

By stepping through the neighborhoods of the vertices of�

we can easily check in lineartime whether � has property �:1 4 � or �:1 < � . So it remains to show the correctness of thealgorithm.

Claim 1. If �*� is a set in � � � �such that � �� then � � �>= � � � �*� �>= � � for all meta–

vertices = of�

.

By the rules of algorithm MARK we immediately conclude � � �>= � � � � � � �>= � � for allmeta–vertices = of

�. Thus � � � � �� settles the proof.

In particular, if � � is < –optimal then � �>= � �� >= � � � , and, if �*� �>= � � � then� �>= � � � .


Claim 2. If � is not �:1 4 � then there are no �:1 4 � –sets in � � � �.

Assume for the contrary that �� is a �:1 4 � –set in � � � �. We will show that then � is a

�:1 4 � –set too. We may choose �� such that it contains a maximum number of � –setsof meta–vertices in

�. Further let = be an arbitrary meta–vertex in

�.

By Claim 1 we may assume that � � is not 2–optimal and � � �>= � � � .If �*� �>= � � � then � � is

4–optimal by Claim 1. Since �� is a �:1 4 � –set there must

be a neighbor ? of = in� " � such that �*� � ? � � � � �� . Therefore � �>= � � �

by rule � � < � .Now let �*� �>= � � � � � � � � � . We will show � �>= � � � � � � � � � too. Supposing� �>= � � � rule � � < � of algorithm MARK must be applied. Hence, there is a neighbor? in

�such that � � is 1–optimal and � � is not 2–optimal, or � � is optimal. If � � is

optimal then we may replace �� >= �by � � , if � � is 1–optimal we can take �� >= � � �

and �*� � ? � � � yielding a contradiction to the choice of � � .Therefore � �>= � � � � � � � � � . In particular, if �� >= � � � then � �>= � � � .Altogether it follows that � is a �:1 4 � –set — a contradiction.

Claim 3. If � is neither �:1 4 � nor �:1 < � then there are no �:1 < � –sets in � � � �.

First note that by Claim 2 there are no �:1 4 � –sets in � � � �. Now assume for the

contrary that there is a �:1 < � –set �� in � � � �. Obviously, �� >= � � � for all meta–

vertices = in�

. By Claim 1, for each meta–vertex = in�

the set � � is 1–optimal andthe set � � is not 2–optimal. Further, rule � � < � cannot be applied for any meta–vertex= in

�since otherwise we can construct a �:1 4 � –set by replacing � � and � � by � � and

� � .Therefore � �*� — a contradiction.

Claim 2 and 3 settle the proof of Lemma 3.3.9.

�Case 2.

� � � and there are no �:1 4 � – and �:1 < � –sets in � � � �.

If there is a �:1 � � –set � then we reduce�

to a meta–vertex�

with the following setparameters: � � � � , �� not defined and �� , where � � � is amarker.

If � is taken to the dominating set of the rest graph, then we substitute � by anarbitrary vertex of � � �

.

Now consider the case that there are no �:1 � � –, �:1 < � – and no �:1 � � –sets.

Case A.�

contains no meta–vertices.Then � � � � � � � . Since there are no �:1 < � –sets there is at least one vertex� � �

with � � � � B . Let� � be the set of vertices

�of

�such that � � � � 4

,or the empty set if there are no such vertices. If there is a vertex

� � �such

that� � � � � � � 4 � then we reduce

�to�

and define � � � � � � �GF � � �>= � � = �� >= � � �>= � � .


If� � � � � � � 4 � for each

� � �then, in particular,

� � contains two nonadjacentvertices. We reduce

�to a meta–vertex

�with the following set parameters:

�� , �� not defined, � � � � � � , �� , � � � � B . Let �� be a

dominating set in the rest graph. Then

� �� " � � � �

� �� ' � � � � = � � � �� " � � 4 � = � � �� = �@?H� � � �� " � � <�� = � � �@?C�N � � �

is a dominating set in � .

Case B.�

contains meta–vertices.

Then we reduce�

to a meta–vertex�

. Note that each minimum set–union of theset parameters of the meta–vertices consists only of � – and � –sets. Further notethat there is at least one meta–vertex = in

�such that � �>= � � � .

Let �� be the set obtained from � by replacing any set � � by � � provided� � � � � � � � . Among all sets � � � � � �

with � � � � choose a set � suchthat the union of all � –sets of �� is maximal. Define � � � �� and

� � � � �

� 4 � �/= � � " � � ��!�>= � � � �� GF � � �>= � �� -� � otherwise $

If there are sets � �� and � � � � " �

such that � � � � � � � . 4 ,� � � �

dominates�

and � � � � � �#" � � then let �� and

�� , �� be undefined. Note� � � � � � 4 .

Else, if there is a set � � � � � �such that � � � �+. 4

, � dominates�

and � " � � then let =+� �be an arbitrary vertex. We define �� ,

�� = � , �� = � .Otherwise, let = � �

be an arbitrary vertex. We define � � � �� = � ,�� , �� is undefined, where � � � is a marker. If � is takento the dominating set of the reduced graph, then we substitute � by an arbitraryvertex of � � �

in � .

Lemma 3.3.10 ([109]) Case 2 can be handled in linear time.

Proof. First recall, that property �:1 � � can be tested in linear time. Further, in Case A alldecisions can be made by stepping through the neighborhoods of the vertices of

�.

So let us consider Case B. It is easy to see that each set � � � � � �with � � � �

contains all 2–optimal � –sets of�

and a remaining collection of 1–optimal � – and � –sets.Thus, applying algorithm MARK gives the desired set �� . Now we check whether there isa vertex = � � " � � such that � � � � = � dominates

�. If there is such a vertex then define

� � � �� and � � � = � . Otherwise, by considering the meta–vertices � for which ��!�� is either � � or an 1–optimal � � , we check whether we can replace the 1–optimal set ��!�� by � � . If there is such a meta–vertex � then define � � � and � � � � �� ' ��!�� H� � � .


All these checks can be easily done in linear time by stepping through the neighborhoodsof the vertices of

�. So the case that we have sets � � � �

such that � � � � � � � . 4 ,� � � �

dominates�

and � � � � � � " � � is handled.

If we do not find such sets � � , � then we check whether we can replace one 2–optimal

set � � by an 1–optimal set � � in order to get a dominating set for�

. Clearly, this is onlypossible if there is exactly one 2–optimal � –set in �� . So we have to compute the numberof 2–optimal � –sets (this can be done while running algorithm MARK) and then, if we haveexactly one such set � � , it must be checked whether � � is 1–optimal and, if so, whether� � � ' � � � � � � dominates

�. Again, this can be done in linear time.

�Case 3.

� � � .If there is a set � of � � � �

such that � � � � and � � � � dominates�

then wereduce

�to a vertex

� � , � � � � � , and define � � � � � �. Furthermore, we add

� � � � � � ' � � �� to the set � .Otherwise, we reduce

�to a meta–vertex

�with the following set parameters: � � �

� � � � , �� and �� are undefined, �� where � � � is a marker.If � is taken to the dominating set of the reduced graph, then we substitute � by anarbitrary vertex of � � �

in � .

Summarizing the above results we obtain

Theorem 3.3.11 ([109]) For solving the generalized domination problem for a graph � wecan contract a homogeneous set

�to a vertex or a meta–vertex � and extend a dominating

set of �� to a dominating set of � in linear time in � � � �.

3.3.4 The � –dominating set problem on� ��

� �! �

For (not necessarily defined) sets � ��$�$�$ �� let � �GF � � ��$�$�$ �� be undefined if each set � � ,2 � �54 ��$�$�$ � � � is not defined, otherwise � �GF � � ��$�$�$ �� denote a defined set of� � ��$�$�$ ��

with minimum cardinality.

Theorem 3.3.12 ([109]) The � –dominating set problem on �� can be solved intime � �� .

Proof. Let � � �� be a graph with at least two vertices. By using modulardecomposition we compute proper homogeneous sets

� ��$�$�$ � � � such that� � ��$�$�$ � � � � is

a partition of � and such that reducing each� � to a representing vertex leads to ��

.By Theorem 3.3.11 we can reduce each

� � , 2 � �54 ��$�$�$ � � � , to a vertex or a meta–vertex.

According to Observation 3.3.5 �� has a pendant vertex ? . Let = be the neighbor

of ? in �� . In the following we show that we can delete ? . We consider the following

four cases :


Case 1. = and ? are meta–vertices.

We update the set parameters of = in the following way:

� � � � �GF � � � � � �GF�� GF � � �� GF �� GF � � ��

and

� � � � � � � �GF �� >= � � � �GF �� >= � � < � $

Case 2. = is a meta–vertex and ? is a vertex.

If � � ? � B , i.e. ? is already dominated, then

� � � � �GF � � � �� ?H� � � � � � � �GF �� ?H� � � � � � � � � � � � � � �and

� �>= � � � �>= �

.

Next, if � � ? � � then

� � � � ?H� � � �GF�� ?H� � � �GF�� where � � , � � remain undefined.

Finally, for B � � ? � � 4 we update the set parameters of = in the following way:

� � � � �GF � � � �� ?H� � � � � � � � �

� � � �� GF �� ?H� � � � � ? � �+<�� GF �� ?H� � � � � � ?H� � � � � ? � 4

and

� � � ��

� �>= � . 4 � � � ? � �n.d. �

� �>= � . 4 ; � � ? � $Case 3. ? is a meta–vertex and = is a vertex.

After deleting ? vertex = will be a meta–vertex in the remaining graph with� � � � = � � � �GF�� and � � undefined.

If � �>= � B then � � � � �GF � � � �� and � � remains undefined.

For � �>= � �both sets � � and � � are not defined.

Finally, if B � �>= � � 4 then we define

� � � �

� �GF�� >= � �+<�� >= � 4 � � � �

�n.d. � � �>= � �+<�� >= � 4 $


� �

� ��

K�� "( MK�� ! MK� � ! M�

� ��

�� , � � n.d.,

� � �� , � � �

�

Case 4

� �

� �K�� "( M K� �"( M��

� �

�� , � � n.d.,

� � �� , � � �

� �

�K�� "( M ��

�� , � � �� ,� � n.d.,

� � n.d.

�

Case 2 � �� , � � �� ,� � n.d.,

� � n.d.

�

Case 2

Figure 3.16: An example for the proof of Theorem 3.3.12.

Case 4. = and ? are vertices.

If � � ? � B , i.e. ? is already dominated, then we need not to update any parameter of= .

For � � ? � �we replace = by a meta–vertex with the following set parameters: If

� �>= � �then � � � � = �@?H� , � � , � � and � � are undefined. Otherwise � � � � = �@?H� ,

� � � � ?H� , � � and � � are undefined.

Finally, if B � � ? � � 4 then we define

� �>= � � �� ? � ) 4 � � �>= � B �� GF � � �>= � � � � ? � ) 4 � � � �>= � B*$

Note, that by Lemma 3.3.4 �� is in �� !�� too. If ��

is the onevertex graph with vertex = , then a minimum � –dominating set

�of � is constructed as

follows :

If = is a meta–vertex, then� � � �GF � � � �� . If = is a vertex then we define

� � �provided � �>= � B and � � ; otherwise let

� � � � � = � .If ��

has at least two vertices then, by Observation 3.3.5, �� has at least

one pendant vertex. As long as possible we delete pendant vertices as described in the abovereduction. Let �� be the remaining graph. By Lemma 3.3.4 �3� is in �� too.If �� is the one vertex graph then we are done as before. Otherwise, we repeat the wholeprocedure by starting with modular decomposition of � � . So, in worst case � � � times modulardecomposition ( � �� . � �� time, see [106, 34]) has to be performed.

Altogether this leads to a � �� time algorithm for computing a minimum � –dominating set for any graph in �� !�� .

�


Since obviously �� is closed under adding false twins by Theorem 2.1.2 weget the following

Corollary 3.3.13 ([109]) The total dominating set problem on �� can be solvedin time � �� .

3.3.5 The � –dominating set problem on distance–hereditary graphs

Note that distance–hereditary graphs are a proper subclass of �� . Indeed, the3–fan (a 1 � with a dominating vertex) is an example for a graph in �� which isnot distance–hereditary.

Recall, that distance–hereditary graphs can be generated from a single vertex by a finitenumber of one vertex extensions (cf. [24, 77]), i.e. by attaching pendant vertices or twins.Since a pair of twins is a homogeneous set we can use an one vertex extension sequenceinstead of modular decomposition. Note, that in [77] it is shown that such a sequence can becomputed in linear time. Thus we get

Theorem 3.3.14 ([109]) For a distance–hereditary graph a minimum � –dominating set canbe computed in linear time.

Proof. At first we compute an one vertex extension sequence � using the linear time algo-rithm presented in [77].

Now we process � in reverse order using the update rules of Theorem 3.3.11 for twinsand Theorem 3.3.12 for pendant vertices. Note that algorithm MARK runs in constant timesince the homogeneous sets are pairs of twins, i.e. sets of constant size two.

In order to get linear running time we store the set parameters of meta–vertices in linkedlists. Additionally we store the cardinalities of the set parameters. Hereby undefined sets areconsidered as sets of infinite cardinality.

Since the update of the set parameters is a simple linking of pointers the algorithm runsin linear time.

�Since distance–hereditary graphs are closed under adding false twins (see [24]) by The-

orem 2.1.2 we get the following

Corollary 3.3.15 ([109]) The total dominating set problem on distance–hereditary graphscan be solved in linear time.

3.3.6 Concluding Remarks

It would be interesting if the results of section 3.3.3 can be used to solve the � –dominatingset problem on homogeneously orderable graphs. These graphs were introduced in [13]as a common generalization of dually chordal and distance–hereditary graphs. For the � –dominating clique and the connected � –dominating set problem on this class polynomialtime algorithms are given in [55], the � –dominating set problem is still open.


In homogeneously orderable graphs ( �� ) we have to consider extremal

vertices instead of pendant vertices. Recall, that a vertex � of a graph � is extremal iff � hasa maximum neighbor, i.e. there is a vertex

� �� 4 � such that � � � � 4 � � �� < � .Analogously to Observation 3.3.5 and Lemma 3.3.4 we have

Observation 3.3.16 ([13]) Let � be a homogeneously orderable graph. Then �� has

at least one extremal vertex.

Furthermore,

Lemma 3.3.17 ([13]) Let � be a homogeneously orderable graph.

(1) If�

is a proper homogeneous set in � and � �+� �then the graph ��

ishomogeneously orderable, too.

(2) If � is an extremal vertex in � then ��) � is homogeneously orderable, too.

Thus, by point � 4 � of the preceding Lemma and by Theorem 3.3.11, it remains to considerthe removal of extremal vertices.

Appendix A

Complexity of Domination Problems

In this chapter we list the complexity of some domination problems on different graph classesbut we declare no claim to completeness.

In the last years at our institute we have developed an information system on graph classinclusions (ISGCI) available via internet under the URL

http://www.informatik.uni-rostock.de/˜gdb/isgci/Isgci.html

to keep an updated knowledge base of graph classes and their inclusions. The user has thepossibility to ask queries about inclusions of classes and to draw inclusion hierarchies forselected classes.

Up to now, the system does not contain any information about the complexity of con-crete graph theoretic problems on graph classes contained in the database. It would be veryinteresting to add known results for well known graph problems such that the user can askqueries. As a starting point the information about MINIMUM DOMINATING SET providedin this chapter can be taken.

Such a system can support research in the following directions:

1. Nearly all published papers are not up–to–date since the referring process takes a longtime. An information system is dynamic if the database is updated regularly.

2. The complexity of a problem can be displayed more clearly in an inclusion hierarchyof by the user selected classes (see Figure A.1 for an example).

3. Such a system can be used to find borders on which the complexity of a problem �changes, i.e. classes � � , � such that � � � and � can be solved efficiently on ��but � remains

��–complete on � . Then, it is interesting to introduce a new class �

between � � , � , i.e. � � � � , such that � can be solved efficiently on � , too.

4. A user can search for open problems. Hereby, maybe results for close-by classes canbe generalized.

51

52 APPENDIX A. COMPLEXITY OF DOMINATION PROBLEMS

P -reducible4

cograph

P -sparse4

AT-free

AT-free ∩ claw-freeco-comparability HHD-free

distance-hereditary

weak bipolarizable

bipartite

chordal bipartite

bipartite ∩ distance-hereditary

tree

block

cactus

chordal

doubly chordal

split undirected path

circle

outerplanar permutation

circular arc

interval

claw-free

line

trapezoid

superfragile threshold

comparability

directed path

ptolemaic

strongly chordal

dually chordal

homogeneously orderable

parity

perfect

quasi-parity

planar

weakly chordal

Figure A.1: The complexity of the minimum dominating set problem on some selected graphclasses. Hereby, a filled (resp. unfilled) ellipse indicates that the problem is

��-complete

(resp. can be solved in polynomial time) and an unfilled box indicates that the complexitystatus is open. Edges from a class on which the problem remains

��-complete to a class

where the problem can be solved in polynomial time are drawn in bold style.

A.1. ABBREVIATIONS FOR PROBLEMS 53

A.1 Abbreviations for problems

Abbreviation Problem

DS Minimum cardinality dominating set problem

RDS Minimum cardinality � –dominating set problem

PDS Minimum cardinality perfect dominating set problem

DC–ex Minimum cardinality dominating clique problem (Existence)

DC–comp Minimum cardinality dominating clique problem (Computation)

RDC–ex Minimum cardinality � –dominating clique problem (Existence)

RDC–comp Minimum cardinality � –dominating clique problem (Computation)

IDS Minimum cardinality independent dominating set problem

IRDS Minimum cardinality independent � –dominating set problem

IPDS Minimum cardinality independent perfect dominating set problem

TDS Minimum cardinality total dominating set problem

TRDS Minimum cardinality total � –dominating set problem

TPDS Minimum cardinality total perfect dominating set problem

CDS Minimum cardinality connected dominating set problem

CRDS Minimum cardinality connected � –dominating set problem

CPDS Minimum cardinality connected perfect dominating set problem

WDS Minimum weighted dominating set problem

WPDS Minimum weighted perfect dominating set problem

WDC–ex Minimum weighted dominating clique problem (Existence)

WDC–comp Minimum weighted dominating clique problem (Computation)

WIDS Minimum weighted independent dominating set problem

WIPDS Minimum weighted independent perfect domination problem

WCDS Minimum weighted connected dominating set problem

WCPDS Minimum weighted connected perfect dominating set problem

WTDS Minimum weighted total dominating set problem

WTPDS Minimum weighted total perfect dominating set problem

STEINER Minimum cardinality Steiner Tree problem

A.2 Complexity on graph classes

For definitions, characterizations and recognition of the mentioned graph classes we refer to[23]. For time complexity we use the following abbreviations:

�for polynomial time,

��–c for

��–complete.


A.2.1 1–CUBs

Problem Time complexity References

DS�

[47]

DC–ex�

[47]

DC–comp�

[47]

TDS�

[47]

CDS�

[47]

A.2.2 2–CUBs


DS��

–c [47]

DC–comp��

–c [47]

TDS��

–c [47]

CDS��

–c [47]

A.2.3 AT–free graphs

co–comparability graphs AT–free graphs


DS � � � � � [89]

DC–ex��

–c see co–comparability graphs

IDS � � �� 3. 4 � � [19]

IPDS � � �� 3. 4 � � [19]

TDS � � � � � [89]

CDS � � � � � [26]

� � � . � �if � �� ; � , [44]

WDS��


WCDS��


WTDS��


STEINER � � � � � [26]

A.2. COMPLEXITY ON GRAPH CLASSES 55

A.2.4 bipartite graphs


DS��

–c [49], see planar bipartite graphs

PDS��

–c [129]

IDS��


IPDS��

–c [125, 131]

TDS��

–c [111]

TPDS��

–c [125, 131]

CDS��


CPDS��

–c [125, 131]

WDC–ex � � � . � �[17], trivial since � �� <

WDC–comp � � � . � �[17], trivial since � �� <

STEINER��

–c see planar bipartite graphs

A.2.5 chordal graphs


DS� �

–c see split graphs, undirected path graphs

PDS� �

–c [129]

DC–ex � � � � �[90]

DC–comp� �

–c [17]

RDC–ex � � � � �[51]

IDS � � � . � �[62]

IPDS� �

–c [125, 131]

TDS� �

–c see split graphs

TPDS� �

–c [125, 131]

CDS� �


WIDS� �

–c [29]

� � � . � �[62], for vertex weights in

�0,1 �

WCPDS � � � . � �[38]

STEINER� �


In [90] it is shown that a chordal graph has a dominating clique if and only if � �� .

A.2.6 chordal bipartite graphs

chordal bipartite graphs bipartite graphs



DS��

–c [103]

PDS��

–c [100]

IDS��

–c [53]

TDS�

[53]

TPDS��

–c [120]

CDS��

–c [103]

WDC–ex � � � . � �see bipartite graphs

WDC–comp � � � . � �see bipartite graphs

STEINER��

–c [103]

A.2.7 circle graphs


DS��

–c [84]

DC–comp � � � � �[84]

TDS��

–c [84]

CDS��

–c [84]

A.2.8 circular–arc graphs

In [101] a linear time recognition algorithm for circular–arc graphs is given. A circular–arcmodel for a circular–arc graph can be computed in the same time bound.


DS � � � �[80], given a circular–arc model

WDS � � � . � �[30], given a circular–arc model

WIDS � � � . � �[30], given a circular–arc model

WCDS � � � . � �[30], given a circular–arc model

WTDS � � � . � �[30], given a circular–arc model


A.2.9 claw–free AT–free graphs


DS � � � . � �[73]

IDS � � � . � �[73]

IPDS � � �� 3. 4 � � see AT–free graphs

TDS � � � � � see AT–free graphs

CDS � � � � � see AT–free graphs

STEINER � � � � � see AT–free graphs

For AT–free graphs with � �� ; � problem CDS can be solved in linear time (see [44]).

A.2.10 co–comparability graphs

co–comparability graphs AT–free

For a given co–comparability graph a co–comparability ordering can be computed in lineartime (see [107]).


DS � � � � �[18], given a co–comparability ordering

DC–ex��

–c [91]

TDS � � � � �[18], given a co–comparability ordering

CDS � � � � �[18], given a co–comparability ordering

WDS��

–c [29]

WIDS � � �� [18]

WIPDS � � � �

[29], given a co–comparability ordering

WCDS��

–c [29]

WTDS��

–c [29]

STEINER�

[91]

A.2.11 comparability graphs

bipartite graphs comparability graphs



DS��

–c see bipartite graphs

PDS��


DC–ex � � � � �[17]

DC–comp � � � � �[17]

IDS��


IPDS��


TDS��


TPDS��


CDS��


CPDS��


STEINER��

–c [67]

A.2.12 convex bipartite


DS � � � �

[14]

IDS � � � �

[14]

TDS � � � �

see DS, [92]

CDS � � � � � [53]

A.2.13 convex–round graphs


DS � � � � � [14]

IDS � � � � � [14]

TDS � � � �[14], given a convex–round enumeration

A.2.14 distance–hereditary graphs

In [68] it is shown that the clique width of distance–hereditary graphs is bounded by threeand that a � –expression can be computed in the same time bound.



RDS � � � . � �Theorem 3.3.14, [109]

RDC–ex � � � . � �[50]

RDC–comp � � � . � �[50]

CRDS � � � . � �[10]

WDS � � � . � �see graph classes of bounded clique width

WPDS � � � . � �see graph classes of bounded clique width

WDC–ex � � � . � �see graph classes of bounded clique width

WDC–comp � � � . � �see graph classes of bounded clique width

WIDS � � � . � �see graph classes of bounded clique width

WIPDS � � � . � �see graph classes of bounded clique width

WCDS � � � . � �[126], see graph classes of bounded clique width

WCPDS � � � . � �see graph classes of bounded clique width

WTDS � � � . � �see graph classes of bounded clique width

WTPDS � � � . � �see graph classes of bounded clique width

STEINER � � � . � �[10], see graph classes of bounded clique width

A.2.15 DSP–graphs (graphs having a dominating shortest path)

co–comparability graphs AT–free graphs DSP–graphs


DS � � � � � [89]

DC–ex��


TDS � � � � � [89]

WDS��


WCDS��


WTDS��


A.2.16 doubly chordal graphs

doubly chordal graphs

chordal graphs"

dually chordal graphs



RDS � � � . � �see dually chordal graphs

RDC–ex � � � . � �see dually chordal graphs

RDC–comp � � � . � �see dually chordal graphs

IDS � � � . � �see chordal graphs

CRDS � � � . � �[9]

WCPDS � � � . � �see chordal graphs

A.2.17 dually chordal graphs


RDS � � � . � �[9]

RDC–ex � � � . � �[51]

RDC–comp � � � . � �[51]

IDS� �

–c [9]

TDS � � � . � �see DS, [92]

CRDS � � � . � � �� [9]

A.2.18 graph classes of bounded clique width

Let � be a graph class of bounded clique width such that for every graph � � � a � –expression can be computed in � ��

time.


WDS � �� Theorem 3.1.3 and 3.1.5

WPDS � �� Theorem 3.1.3 and 3.1.5

WDC–ex � �� Theorem 3.1.3 and 3.1.5

WDC–comp � �� Theorem 3.1.3 and 3.1.5

WIDS � �� Theorem 3.1.3 and 3.1.5

WIPDS � �� Theorem 3.1.3 and 3.1.5

WCDS � �� Theorem 3.1.3 and 3.1.5

WCPDS � �� Theorem 3.1.3 and 3.1.5

WTDS � �� Theorem 3.1.3 and 3.1.5

WTPDS � �� Theorem 3.1.3 and 3.1.5

STEINER � �� Theorem 3.1.3 and 3.1.5


A.2.19 homogeneously orderable graphs

In [13] an � � � � � recognition algorithm for homogeneously orderable graphs is given. A�

–extremal ordering for a homogeneously orderable graph can be computed in the same timebound.


RDC–ex � � � �

[55], given an�

–extremal ordering

RDC–comp � � � �

[55], given an�


IDS��

–c see dually–chordal graphs

CRDS � � � �

[55], given an�


STEINER � �� [13], given an

�–extremal ordering

A.2.20 interval graphs

In [21] a linear time algorithm for recognizing interval graphs is given. In the same timebound an interval model can be constructed using 1 � –trees.


RDS � � � . � �see dually chordal graphs

RDC–ex � � � . � �see dually chordal graphs

RDC–comp � � � . � �see dually chordal graphs

TDS � � � . � �[83]

WDS � � � �[30], given an interval model

WPDS � � � . � �[39]

WIDS � � � �[30], given an interval model

WIPDS � � � . � �[39]

WCDS � � � �[30], given an interval model

WCPDS � � � . � �[39]

WTDS � � �� , � � , � �[30], given an interval model

� � � . � �[114]

WTPDS � � � . � �[39]

STEINER � � � �

see strongly chordal graphs

A.2.21 line graphs

The edge version of domination can be thought of as the vertex version of the problemapplied to line graphs.



DS��

–c [128]

PDS��

–c [102]

IDS��

–c [128]

TDS��

–c [102]

A.2.22 partial � –trees (for bounded � )

For a given � partial � –trees are exactly the graphs having tree width (� � � ) at most � . Since

� � �� <�� .�� 7 � . 4 (see [43])

graphs of bounded tree width are of bounded clique width, too.

In [2] it is shown that for a given partial � –tree an embedding in a � –tree can be found inpolynomial time.


DS � � � . � �[5], given a � –tree embedding

DC–comp�

[4]

IDS�

[4]

TDS�

[4]

CDS�

[4]

WDS�

see graph classes of bounded clique width

WPDS�


WDC–ex�


WDC–comp�


WIDS�


WIPDS�


WCDS�


WCPDS�


WTDS�


WTPDS�


STEINER�


A.2.23 permutation graphs

permutation graphs trapezoid graphs


In [107] a linear time recognition algorithm for permutation graphs is given. A permuta-tion diagram for a permutation graph can be computed in the same time bound.


DS � � � �[35], given a permutation diagram

DC–ex � � � �[16, 17, 81], given a permutation diagram

DC–comp � � � �[17, 81], given a permutation diagram

IDS � � �� , � �[3], given a permutation diagram

TDS � � � �see DS, [92], given a permutation diagram

CDS � � � �[81], given a permutation diagram

WDS � � � . � �[113]

WPDS � � �� , � �see trapezoid graphs

WDC–ex � � �� , � �[119], given a permutation diagram

WDC–comp � � �� , � �[119], given a permutation diagram

WIDS � � � . � �[99]

WCDS � � � . � �[94]

STEINER � � �� , � �[81], given a permutation diagram

A.2.24 planar graphs


DS��


PDS��

–c [86]

IDS��


CDS��


STEINER��


A.2.25 planar bipartite graphs


DS��

–c [132]

IDS��

–c [132]

CDS��

–c [124]

WDC–ex � � � . � �see bipartite graphs

WDC–comp � � � . � �see bipartite graphs

STEINER��

–c [124]


In [132] it is shown that DS and IDS are��

–c for the subclass�

of planar bipartitegraphs. Hereby, � belongs to

�if the following conditions are fulfilled:

� � is planar,

� � is bipartite,

� � has maximum degree 3,

� � has the girth�� , where � is fixed.

A.2.26 � –polygon graphs for fixed ��In [59] an � � � % � � time algorithm recognizing � –polygon graphs (for fixed �� ) is given.


DS � � � � % � 7 � � [59, 58], given a � –polygon representation

DC–comp�

[84]

IDS � � < % � � � % � � � [58], given a � –polygon representation

TDS � � � � % � 7 � � see DS, [92], given a � –polygon representation

CDS � � � � % � 7 � � [58], given a � –polygon representation

A.2.27 series–parallel graphs = partial 2–trees


DS � � � �[93]

IDS � � � [112], see partial � –trees

TDS � � � �[112], see partial � –trees

WDS�

see partial � –trees

WPDS � � � �[125, 130]

WDC–ex�


WDC–comp�


WIDS�


WIPDS � � � �[125]

WCDS � � � �[124]

WCPDS � � � �[125]

WTDS�


WTPDS � � � �[125]

STEINER � � � �[123]


A.2.28 split graphs

split graphs chordal graphs


DS��

–c [45]

DC–comp��

–c see chordal

RDC–ex � � � � �see chordal

IDS � � � . � �see chordal

TDS��

–c [47]

CDS��

–c [124]

WPDS � � � . � �[38]

WIPDS � � � . � �[38]

WCPDS � � � . � �[38]

WTPDS � � � . � �[38]

STEINER��

–c [124]

A.2.29 strongly chordal

strongly chordal

hereditary dually chordal dually chordal

For a given graph � one can find a simple (resp. strong) elimination ordering of � in� � � �GF � � � � , � � �

� � (resp. � � � � � ) time (see [23]).


RDS � � � . � �see dually chordal

� � � . � �[30], given a simple elimination ordering

RDC–ex � � � . � �see dually chordal

RDC–comp � � � . � �see dually chordal

TDS � � � . � �see dually chordal

� � � . � �[28], given a simple elimination ordering

CRDS � � � . � �[30], given a simple elimination ordering

WDS � � � . � �[63], given a strong elimination ordering

WIDS � � � . � �[63], given a strong elimination ordering


STEINER � � � �

[124], given a strong or simple elimination ordering


A.2.30 trapezoid graphs

trapezoid graphs co–comparability graphs


DC–ex � � � . � �[85], given a trapezoid order

DC–comp � � � . � �[85], given a trapezoid order

TDS � � � � �[95]

CDS � � � �[85], given a trapezoid diagram

WDS � � � � �[95]

WPDS � � �� , � �[96], given a trapezoid diagram

WDC–ex � � � � � , � � �[1]

WDC–comp � � � � � , � � �[1]

WIDS � � �� , � �[96], given a trapezoid diagram

WIPDS � � � �

see co–comparability graphs

WCDS � � � . �� , � �[117], given a trapezoid diagram

STEINER�

see co–comparability graphs

A.2.31 trees

trees distance–hereditary graphs



DS � � � �[33]

RDS � � � �[116]

RDC–ex � � � �trivial

RDC–comp � � � �trivial

CRDS � � � �see distance–hereditary graphs

WDS � � � �[110]

WPDS � � � �[129]

WDC–ex � � � �trivial

WDC–comp � � � �trivial

WIDS � � � �see distance–hereditary graphs

WIPDS � � � �[125]

WCDS � � � �see distance–hereditary graphs

WCPDS � � � �[125]

WTDS � � � �see distance–hereditary graphs

WTPDS � � � �[125]

STEINER � � � �trivial

A.2.32 undirected path graphs

undirected path graphs chordal graphs


DS��

–c [15]

DC–ex�

[88]

DC–comp�

[88]

RDC–ex � � � � �see chordal graphs

IDS � � � . � �see chordal graphs

TDS��

–c [98]

CDS��

–c [47]


A.2.33 weakly chordal graphs

split graphs chordal graphs weakly chordal graphs



DS��


PDS��

–c see chordal graphs

DC–ex��

–c [17]

DC–comp��

–c [17]

IPDS��


TDS��


TPDS��


CDS��


WIDS��


STEINER��


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78 BIBLIOGRAPHY

Index

� �� , 31

� � � � �� , 10

�� , 5

� ( , 4�-! , 4� , 10�� , 11 ��

, 10�, 13

�� , 18� �� , 120 ��

, 32-color-width, 18

abstract dominating set, 39

clique, 4clique width, 10�� , 5�� , 18complete graph, 4complete set, 4component

connected, 4connected, 4connected component, 4 � � ��

, 10cycle, 3

chordless, 3length, 3

� �� , 7

� �� , 4� �� E �

, 3� �>= �@? � , 3� �-,/. �� , 3degree, 3� ��

, 4

diameter, 4disc, 4disconnected, 4distance, 3distance–hereditary, 5domain, 11dominating set, 5

abstract, 39edge, 26minimum, 5minimum weighted , 6perfect, 6

domination number, 5duplex graph, 7

� �� , 3

� �� , 4eccentricity, 4edge, 3edge dominating set, 26edgeless graph, 4extremal, 32

� �� , 12� –free, 4� -free, 4�� $�$�$ �� –free, 4false twin, 5first–order logic, 11FOL, 11formula

in first–order logic, 11FT, 5

� � �� , 3generation sequence, 32graph

clique width, 10

79

80 INDEX

complete, 4connected, 4disconnected, 4duplex, 7edgeless, 4� -free, 4induced subgraph, 3prime, 4, 30subgraph, 3

graphsjoin of, 4union of, 4

� � , 35hereditary, 4��

, 30homogeneous extension, 30homogeneous reduction, 30homogeneous set, 4, 29

nontrivial, 4, 30overlap–free, 30proper, 30trivial, 4

homogeneous setsoverlap, 30

�� , 30

independent set, 4induced by, 3ISGCI, 51isometric subgraph, 5

join of � and �� , 4

�� , 5

� �� , 26

line graph, 26LinEMSOL( � �� ), 13logic

first–order, 11monadic second–order, 12second–order, 12

� , 3maximum neighbor, 32

� �GF�� $�$�$ �� , 45MINIMUM CONNECTED DOMINATING

SET, 6MINIMUM DOMINATING CLIQUE, 6MINIMUM DOMINATING SET, 5MINIMUM EDGE DOMINATING SET, 26MINIMUM INDEPENDENT DOMINATING

SET, 6MINIMUM TOTAL DOMINATING SET, 6minimum weighted dominating set, 6MN, 32module tree, 30monadic second–order logic, 12MSOL, 12

� , 3� � , 4 � �

, 3 �� , 3�� , 3�� , 3neighbor, 3neighborhood, 3� –th, 3 % �� , 3

nontrivial, 30

� � � �, 3

� –expression, 10� –graph, 10path, 3

chordless, 3length, 3

pendant vertex, 5perfect dominating set, 61�� , 5��

, 30prime, 4prime graph, 30PV, 5

� -dominating set, 5red column-alternating vertex, 18red row-alternating vertex, 18reduction sequence, 32

INDEX 81

relation symbol, 11arity, 11

�� , 18

� .�� , 12�� , 12second–order logic, 12sequence

generation, 32reduction, 32

setcomplete, 4dominating, 5homogeneous, 4, 29independent, 4stable, 4total, 6

SOL, 12stable set, 4Steiner set, 6STEINER TREE, 6structure, 11subgraph, 3

induced, 3isometric, 5

� � �� , 30

total, 6tree, 4true twin, 5TT, 5� � , 4twin

false, 5true, 5

� � , 3union of � and �� , 4

� �� , 3

variable, 11vertex

degree, 3extremal, 32neighbor, 3neighborhood, 3

82 INDEX

Thesen

1. Viele Probleme, die sich mit dem strategischen Plazieren von irgendwelchen Objek-ten in einem Netzwerk beschaftigen, lassen sich als Dominationsprobleme in gewissenGraphen beschreiben. Im einfachsten Fall, dem sogenannten Problem MINIMUM DO-MINATING SET sucht man zu einem gegebenen Graphen � � � ��

eine kleinsteTeilmenge � von � so, daß fur jeden Knoten � aus � gilt: � � � oder � hat einenNachbarn in � . Durch die Betrachtung von konkreten Anwendungen wurden mehrereVarianten dieses Problems formuliert und algorithmisch untersucht. Zu erwahnen isthierbei u.a. das Steinerbaumproblem (STEINER TREE), ein Spezialfall des ProblemsMINIMUM CONNECTED � –DOMINATING SET, welches im VLSI–Design verwendetwird.

2. Leider ist MINIMUM DOMINATING SET, wie auch fast alle Varianten hiervon, ein��

–vollstandiges Problem. Daher ist es interessant, die Komplexitat dieses Problemsfur spezielle Graphenklassen zu untersuchen. Insbesondere mochte man fur moglichstgroße Klassen effiziente Algorithmen entwickeln.

3. Graphenklassen mit beschrankter Cliquenweite besitzen in der algorithmischen Gra-phentheorie eine große Bedeutung. Fur solche Klassen existiert ein gemeinsamer Zu-gang, mit dem sich viele Graphenprobleme (sozusagen gleichzeitig) effizient losenlassen. Voraussetzung hierfur ist u.a., daß sich das gegebene Problem in monadischerLogik zweiter Ordung beschreiben laßt. MINIMUM DOMINATING SET ist ein solchesProblem, das Problem MINIMUM � –DOMINATING SET dagegen nicht.

4. Bis auf vier ausnahmen laßt sich fur jede Graphenklasse, die durch zwei verbotenevier–Knoten Graphen definiert ist, angeben, ob die Cliquenweite durch eine Konstantebeschrankt ist, oder nicht (siehe Figure 3.1 in der Dissertation). Fur die folgendenKlassen ist dieses Problem noch offen:

� � � �� !� < � �-freie Graphen,

� � �� , co–diamond)-freie Graphen,� � � co– � � � � �� -freie Graphen,� � � co– � � � �� , diamond)-freie Graphen.

5. Es sei � eine Menge von Graphen, die hochstens vier Knoten enthalten. Fur � –freieGraphen laßt sich MINIMUM DOMINATING SET entweder in Polynomialzeit losen,

83

84 THESEN

oder � ist eine Oberklasse der folgenden Klassen und MINIMUM DOMINATING SET

ist��

–vollstandig:

� � � � � < � �-freie Graphen,

� (claw, � � , diamond, � � )-freie Graphen,� ( � � , � � )-freie Graphen.

6. Die Graphenklasse �� ist eine Teilklasse der homogen geordneten Gra-phen, welche die distanz–erblichen Graphen enthalt, und unbeschrankte Cliquenweitebesitzt.

(a) Die Erkennung, ob ein Graph Element der Klasse �� ist, laßt sich inZeit � �� durchfuhren.

(b) Die Probleme MINIMUM � –DOMINATING SET und TOTAL DOMINATING SET

lassen sich fur Graphen aus �� in Zeit � �� losen. Fur dieTeilklasse der distanz–erblichen Graphen verringert sich die Zeitschranke aufLinearzeit.

7. In den letzten Jahren wurde am Institut fur Theoretische Informatik der UniversitatRostock ein Informationssystem uber Graphenklasseninklusionen (ISGCI) entwickelt,welches im Internet unter der URL

http://www.informatik.uni-rostock.de/˜gdb/isgci/Isgci.html

erreichbar ist. Die im Appendix A der Dissertation angegebene Ubersicht der Komple-xiat von 27 Dominationsproblemen fur verschiedene Graphenklassen kann als Aus-gangspunkt dienen, dieses System mit Informationen uber graphentheoretische Pro-bleme zu erweitern.

Tabellarischer Lebenslauf

Angaben zur PersonName Thomas Szymczak

Geburtsdatum, Geburtsort 13.03.2001, Dinslaken (NRW)

Staatsangehorigkeit deutsch

Familienstand ledig

Schulausbildung / Studium1977-1981 Grundschule in Dinslaken-Hiesfeld

1981-1990 Otto-Hahn-Gymnasium in Dinslaken

Abschluß: Allgemeine Hochschulreife

1991-1995 Studium der Mathematik mit Nebenfach Informatik an der Gerhard-Mercator-Universitat – Gesamthochschule Duisburg

Abschluß: Diplom-Mathematiker

Grundwehrdienst1990-1991 Ausbildung als Kfz- und Panzerschlosser in Varel und Wesel

Weitere Tatigkeiten1993-1994 Entwicklung eines Ubungsbuches zur Analysis fur den Vieweg Verlag

1995 DFG-Forschungsstudent bei Prof. Dr. A. Brandstadt

Thema: Parallele und sequentielle Algorithmen fur Probleme auf Hy-pergraphen und Graphen mit Baumeigenschaften

seit 1995 Wissenschaftlicher Assistent (C1) am Institut fur Theoretische Informa-tik an der Universitat Rostock

1997 Einmonatiger Forschungsaufenthalt am Wissenschaftlichen Zentrumder IBM in Heidelberg

Thema: Beschreibung eines Vehicle Routing Problems

85

86 TABELLARISCHER LEBENSLAUF

Erklarungen gemaß�3, Absatz 1, Punkt

7 und 8 der Promotionsordnung

1. Hiermit erklare ich, daß ich die eingereichte Dissertation selbstandig und ohne fremdeHilfe verfasst, andere als die von mir angegebenen Quellen und Hilfsmittel nicht be-nutzt und die den benutzten Werken wortlich oder inhaltlich entnommenen Stellen alssolche kenntlich gemacht habe.

2. Hiermit erklare ich, daß ich mich weder an der Universitat Rostock noch an eineranderen Universitat um den Doktorgrad beworben habe.

Rostock, 07.12.2001

87

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