On Computational Aspects of Greedy Partitioning of Graphs

Abstract

In this paper we consider a variant of graph partitioning consisting in partitioning the vertex set of a graph into the minimum number of sets such that each of them induces a graph in hereditary class of graphs P (the problem is also known as P-coloring). We focus on the computational complexity of several problems related too greedy partitioning. In particular, we show that given a graph G and an integer k deciding if the greedy algorithm outputs P-coloring with at least k colors is NPcomplete if P is a class of K p -free graphs with p ≥ 3. On the other hand we give a polynomial-time algorithm when k is fixed and the family of minimal forbidden graphs defining the class P is finite. We also prove coNP-completeness of deciding if for a given graph G and an integer t ≥ 0 the difference between the largest number of colors used by the greedy algorithm and the minimum number of colors required in any P-coloring of G is bounded by t. In view of computational hardness, we present new Brooks-type bound on the largest number of colors used by the greedy P-coloring algorithm.

Computational aspects of greedy partitioning of graphs

“…An extended abstract of a preliminary version of this paper has been presented at 11th International Workshop on Frontiers in Algorithmics, FAW 2017 (see Borowiecki 2017).…”