Computational aspects of greedy partitioning of graphs

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 ℕℙ -complete 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.