On Computational Aspects of Greedy Partitioning of Graphs

Abstract

In this paper we consider a problem of graph \({\mathcal P}\)-coloring consisting in partitioning the vertex set of a graph such that each of the resulting sets induces a graph in a given additive, hereditary class of graphs \({\mathcal P}\). We focus on partitions generated by the greedy algorithm. In particular, we show that given a graph G and an integer k deciding if the greedy algorithm outputs a \({\mathcal P}\)-coloring with a least k colors is \(\mathbb {NP}\)-complete for an infinite number of classes \({\mathcal P}\). On the other hand we get a polynomial-time certifying algorithm if k is fixed and the family of minimal forbidden graphs defining the class \({\mathcal P}\) is finite. We also prove \(\mathrm{co}\mathbb {NP}\)-completeness of the problem of deciding whether for a given graph G the difference between the largest number of colors used by the greedy algorithm and the minimum number of colors required in any \({\mathcal P}\)-coloring of G is bounded by a given constant. A new Brooks-type bound on the largest number of colors used by the greedy \({\mathcal P}\)-coloring algorithm is given.