Graph Clustering : Complexity , Sequential and

In this thesis we study graph clustering on two well known families of graphs that arise in many applications, namely bipartite and chordal graphs. We study two speci c types of clustering problems. On the one hand we seek a partition of the vertex set of a graph into a bounded number of sets so that a prespeci ed function determined by the diameters of the induced subgraphs is minimized. On the other hand we seek one subset of the vertex set that has a xed number of vertices and induces the maximum possible number of edges. The rst problem type constitutes the main theme of this thesis. We de ne a class of functions on partitions of vertices, which we call monotone diameter functions, that will enable us to group many problems into one general problem. Examples of these problems are the following: partition the vertices of a graph into a limited number of subgraphs of bounded diameter, and partition the vertices of a graph into a limited number of subgraphs so that the sum of the diameters of the subgraphs is bounded. We prove that the problem of partitioning the vertices of a graph into subgraphs of bounded diameter is NP-complete on bipartite and chordal graphs. In contrast with these negative results, we give linear time sequential algorithms for the same problem on bipartite permutation and interval graphs, the rst being a subset of bipartite graphs and the second being a subset of chordal graphs. We also present e cient parallel algorithms for the general problem on biconvex and interval graphs. During the course of studying this problem on biconvex graphs, we prove that biconvex graphs have a certain ordering of the vertices, which we call a biconvex straight ordering, that has promising algorithmic potential. We present an e cient parallel algorithm for constructing a biconvex straight ordering for any biconvex graph. We also study the problem of nding a cluster of bounded size with the maximum possible number of edges, and present an e cient parallel algorithm for solving this problem on a class of unit interval graphs. Acknowledgements To God I owe everything. To everyone that helped make the completion of this thesis possible I o er my thanks. My supervisor, Lorna Stewart, provided me with expert guidance, many hours of discussion, and nancial support to complete the thesis and to attend conferences. I am grateful for her time, e ort, patience, and support. I am thankful to the other members of my committee: Stephen Hedetniemi, Jim Hoover, Joseph Culberson, and Andy Liu. Special thanks to Stephen Hedetniemi for suggesting many thoughtful improvements, to Jim Hoover for the time he spent with me discussing some of the parallel algorithms in this thesis, to Joseph Culberson for carefully reading the thesis, and to Andy Liu for pointing out a shorter proof for one of the theorems. To my parents, I am indebted with gratitude for their in nite support throughout all stages of the preparation of this thesis, and throughout all stages of my life. I am thankful to my husband, Ehab, for o ering me invaluable advice on thesis writing and for his patience and support. To my children, Aser and Salma, I am grateful for their love, smiles, laughter, and hugs.