Partitions of graphs into cographs

Abstract

Cographs form the minimal family of graphs containing K 1 that is closed with respect to complementation and disjoint union. We discuss vertex partitions of graphs into the smallest number of cographs. We introduce a new parameter, calling the minimum order of such a partition the c - chromatic number of the graph. We begin by axiomatizing several well-known graphical parameters as motivation for this function. We present several bounds on c -chromatic number in terms of well-known expressions. We show that if a graph is triangle-free, then its chromatic number is bounded between the c -chromatic number and twice this number. We show that both bounds are sharp for graphs with arbitrarily high girth. This provides an alternative proof to a result by Broere and Mynhardt; namely, there exist triangle-free graphs with arbitrarily large c -chromatic numbers. We show that any planar graph with girth at least 11 has a c -chromatic number at most two. We close with several remarks on computational complexity. In particular, we show that computing the c -chromatic number is NP-complete for planar graphs.