Partitions of Graphs

Abstract

Many difficult optimization problems on graphs become tractable when restricted to some classes of graphs, usually to hereditary classes. A large part of these problems can be expressed in the vertex partitioning formalism, i.e., by partitioning of the vertex set of a given graph into subsets \({V }_{1},\ldots,\!{V }_{k}\) called colour classes, satisfying certain constraints either internally or externally, or both internally and externally. These requirements may be conveniently captured by the symmetric k-by-k matrix M. Concepts which are modeled by M-partitions fall naturally into the three types; each is represented in this work by some problem. Any minimal reducible bound for a hereditary property is in some sense the best possible partition. A number of such partitions are given. Clustering is a central optimization problem (among many others) with applications in various disciplines, e.g., computational biology, communications networks, image processing, pattern analysis [41, 53, 60, 57], and numerous other fields. Some new results on k-clustering of graphs are proved. Another type of M-partition is a matching cutset. The main known results on this subject are collected. The last part of this work is devoted to acyclic partitions of graphs where we consider important classes of graphs and their acyclic reducible bounds. For each partition type the complexity of considered problems is given. Also a number of open problems are presented.