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.

Supported by the Polish National Science Center grant DEC-2011/02/A/ST6/00201.