Algorithmic and structural aspects of graph partitioning and related problems

Abstract

Graph partitioning problems enjoy many practical applications as well as algorithmic and theoretical challenges. This motivates the topics of this thesis that is composed of two parts. The first part of the thesis consists of Chapters 2 to 4. In this part, we present results on the complexity and inapproximability of some vertex partitioning problems, and we give approximation algorithms and on-line algorithms for some other vertex partitioning problems. We will start by investigating the inapproximability and complexity of the problems of finding the minimum number of monochromatic cliques and rainbow cycles that, respectively, partition V (G), where the graph G avoids some forbidden induced subgraphs. Secondly, we study the complexity, and develop approximation algorithms and on-line algorithms for injective coloring problems, to be defined later. Finally, we consider the design of a semidefinite programming based approximation algorithm for a variant of the max hypergraph cut problem. The second part of the thesis consists of Chapters 5 to 7. In this part, we turn our attention to structural properties of some problems that are related to matching problems which be regarded as edge partitioning problems. Firstly, we determine the minimum size of a k-extendable bipartite graph and that of an n-factor-critical graph, and we study the problem of determining the minimum size of a k-extendable non-bipartite graph. We solve this problem for k = 1 and k = 2, and we pose a conjecture related to the problem for general k. Secondly, we improve two equivalent structural results due to Woodall and Las Vergnas on the existence of a directed Hamilton cycle in a digraph and the containment of every perfect matching in a Hamilton cycle in a balanced (undirected) bipartite graph, respectively. Finally, we study a generalization of the maximum matching problem called the maximum triangle set problem, in which the aim is to find the maximum number of vertex-disjoint triangles of a given graph. We present a necessary and sufficient condition for augmenting triangle sets, analogous to the well-known augmenting path result for matchings.