THE MAXIMUM INDUCED MATCHING PROBLEM FOR SOME SUBCLASSES OF WEAKLY CHORDAL GRAPHS Thesis Submitted to The College of Arts and Science of the UNIVERSITY OF DAYTON in Partial Fulfillment of the Requirements for The Degree Master of Computer Science by Chandra Mohan Krishnamurthy UNIVERSITY OF DAYTON Dayton, Ohio December, 2009 THE MAXIMUM INDUCED MATCHING PROBLEM FOR SOME SUBCLASSES OF WEAKLY CHORDAL GRAPHS APPROVED BY: Barbara A. Smith, Ph.D. Dale Courte, Ph.D. Committee Member Committee Member Associate Professor, Chair and Associate Professor, Department of Computer Science Department of Computer Science R. Sritharan, Ph.D. Advisor Associate Professor, Department of Computer Science ii c Copyright by Chandra Mohan Krishnamurthy All rights reserved 2009 ABSTRACT THE MAXIMUM INDUCED MATCHING PROBLEM FOR SOME SUBCLASSES OF WEAKLY CHORDAL GRAPHS Name: Krishnamurthy, Chandra Mohan University of Dayton Advisor: Dr. R. Sritharan An induced matching in a graph is a set of edges such that no two edges in the set are joined by any third edge of the graph. An induced matching is maximum (MIM) if the number of edges in it is the largest among all possible induced matchings. It is known that finding the size of MIM in a graph is NP-hard even if the graph is bipartite. It is also known that the size of MIM in a chordal graph or in a weakly chordal graph can be computed in polynomial time. Specifically, the size of MIM can be computed in linear time for a chordal graph and in O(m3) time for a weakly chordal graph. This work demonstrates some algorithms for the maximum induced matching problem with complexity better than O(m3) for some subclasses of weakly chordal graphs. In particular, we show that the maximum induced matching problem can be solved on hhd-free graphs in O(m2) time. Further, for two subclasses of hhd- free graphs, we show that the problem can be solved in better time than O(m2). The classes of graphs that we consider are either more general than chordal graphs or are a restriction of chordal bipartite graphs. iii Dedicated to my parents... Thank you Amma and Daddy. ACKNOWLEDGEMENTS I owe my deepest gratitude to Dr. R. Sritharan, my advisor, for providing me the time and guidance throughout my graduate studies, and for directing this thesis and bringing it to its conclusion with patience and expertise. This thesis would not have been possible without his immense support. It is a pleasure to thank the members of the thesis committee for their invaluable time and help. I would also like to thank the department of computer science and Dr. James P. Buckley for the financial support during the summer term. This thesis is dedicated to my parents who have always believed in me and stood by my side. Encouragement from my family and friends has been a great help towards the completion of this thesis. Further, I am indebted to my close friends for being by my side at all times. Thanks. iv TABLE OF CONTENTS CHAPTER I INTRODUCTION 1 Definitionandnotation............................. 1 BackgroundandMotivation . 2 II HHD-FREE GRAPHS 8 IntroductionandMotivation. 8 Theorem..................................... 8 Implications................................... 22 III A SUBCLASS OF CHORDAL BIPARTITE GRAPHS 24 Introduction................................... 24 Doublylexicalordering............................. 24 Perfect ordering of G∗ ............................. 25 h is in region A (Re = Rh)........................ 30 h is in region E (Re = Rh)........................ 30 h is in region C (Ce = Ch)........................ 30 h isinregionB .............................. 31 h isinregionD .............................. 33 Greedy coloring algorithm for G∗ ....................... 45 v Correctnessofalgorithm2 . 47 Case1................................... 47 Case2................................... 57 IV A SUBCLASS OF HHD-FREE GRAPHS 64 Forbiddenconfiguration . .. .. .. .. .. ... .. .. .. .. .. ... 65 Perfect ordering of G∗ ............................. 66 Case 1: b = d < f. ............................ 68 Case 2: b<d<f. ............................ 70 Coloring of G∗ .................................. 75 Case1................................... 76 Case2................................... 78 V SUMMARY AND CONCLUSIONS 80 Epilogue..................................... 80 BIBLIOGRAPHY 82 vi LIST OF FIGURES I.1 hole,houseanddomino.......................... 2 I.2 Graph ................................... 3 I.3 chordalandweaklychordalgraphs . 5 I.4 HHD-freegraphs ............................. 6 II.1 house in G∗ ................................ 9 II.2 Case1oflemmaII.0.2 .......................... 10 II.3 Case2oflemmaII.0.2 .......................... 11 II.4 Case2oflemmaII.0.2 .......................... 11 II.5 Case3oflemmaII.0.2 .......................... 12 II.6 Case3oflemmaII.0.2 .......................... 12 II.7 Case3oflemmaII.0.2 .......................... 13 II.8 Case3.1oflemmaII.0.2 ......................... 14 II.9 Case3.2oflemmaII.0.2 ......................... 15 II.10Case3.2oflemmaII.0.2 . 15 II.11Case3.2oflemmaII.0.2 . 16 II.12Case3.2oflemmaII.0.2 . 16 II.13Case3.2oflemmaII.0.2 . 17 II.14Case3.2oflemmaII.0.2 . 17 II.15 Domino in G∗ ............................... 19 II.16 e2,e3,e4,e5 in G .............................. 19 vii II.17Case1oflemmaII.0.3 .......................... 20 II.18Case2oflemmaII.0.3 .......................... 21 II.19Case4oflemmaII.0.3 .......................... 21 III.1 Obstruction in M ............................. 26 III.2 Obstruction in G∗ ............................ 29 III.3 C4 with 3 pendents (F1 )......................... 45 IV.1P...................................... 64 IV.2 F2 ..................................... 64 IV.3 Forbidden configuration in G ....................... 65 IV.4Triple ................................... 67 IV.5Anhhd-freegraph ............................ 67 IV.6 Triple in G∗ ................................ 68 IV.7 Order when g ischosen.......................... 69 IV.8 a, b, c and d in G.............................. 69 IV.9 Case 2: b = e ............................... 71 IV.10 Case 2: a = e ............................... 71 IV.11 Case 2.1: {a, b} ∩ {e, f} = ∅ and bf ∈ E ................ 72 viii CHAPTER I INTRODUCTION This chapter introduces and provides an overview of the work done in this thesis. The first section of this chapter contains some definitions and notation used and followed throughout this text. The next section provides the background for the chosen topic for this thesis. It also motivates the reasoning behind the choice of problem and techniques used to solve it. The last section provides an overview of the remaining chapters of the thesis. Definition and notation Finite set of vertices (V ) and finite set of edges (E) together form a graph, G = (V,E). Two vertices x and y are said to be adjacent if xy ∈ E. The cardinality of the set E is referred to as m and the size of the set V is referred to as n. A subgraph H =(V ′,E′) of G =(V,E) is said to be an induced subgraph if for any pair of vertices (x, y) such that x ∈ V ′ and y ∈ V ′, xy ∈ E′ if and only if xy ∈ E. Complement of a graph G is the graph G on the same vertices such that two vertices of G are adjacent if and only if they are not adjacent in G. The neighborhood of a vertex v is the set of all vertices that are adjacent to v, it is denoted by N(v). A chordless path is a sequence (v1v2...vk) of vertices such that every two consecutive vertices in the sequence are adjacent and no other pair of vertices are adjacent. A chordless path 1 2 with k vertices is denoted Pk. If the only other edge present in the sequence is v1vk then it is called an induced cycle. An induced cycle with k vertices is denoted by Ck. An induced cycle of length ≥ 5 (Ck, k ≥ 5) is called a hole. Complement of an induced path on 5 vertices (P5) is called a house.A domino consists of 2 induced cycles of length 4 which share exactly one edge. These graphs are illustrated in the Figure I. e f f b a e a c a d e b d d b c c Hole House Domino Figure I.1: hole, house and domino A clique is a set of vertices that induces a complete graph. Size of a clique is the number of vertices in it. The size of a largest clique in graph G is denoted by ω(G). Chromatic number of G, denoted by χ(G), is the minimum number of colors needed to color vertices of G so that adjacent vertices do not receive the same color. An independent set of a graph is a set of pairwise nonadjacent vertices. An independent set is maximum if it is the largest among all independent sets. A 2K2 is the complement of a C4. Background and Motivation A matching in a graph is a set of edges such that no two of them share an endpoint. An induced matching in a graph is a matching such that no two edges in 3 the matching are joined by any third edge of the graph; that is, an induced matching is a matching which forms an induced subgraph. Sometimes an induced matching is also referred to as a strong matching. Consider the graph shown in the following Figure I.2. f e c d b a Figure I.2: Graph Here, the set {ab, cf, de} is a matching but not an induced matching. On the other hand, The set {cf, de} is an induced matching. The size of an induced matching is the number of edges in the induced matching. An induced matching is maximum if its size is the largest among all possible induced matchings. A largest matching in a graph can be computed in polynomial time [6]. In contrast, Cameron [3] and Stockmeyer and Vazirani [22] showed that finding a max- imum induced matching (MIM) is NP-hard even when the input graph is bipartite. MIM problem refers to the problem of finding the size of a maximum induced match- ing in a graph. There is an immediate connection between the size of an induced matching and the irredundancy number of a graph [11]. On the practical side, the following is an application of induced matching for secure communication channels given by Golumbic and Lewenstein [11]. Consider a bipartite graph G = (X,Y,E). 4 Let X represent the broadcaster vertices and Y represent the receiver vertices.