On the (parameterized) complexity of recognizing well-covered (r, `)-graphs

An (r, `)-partition of a graph G is a partition of its vertex set into r independent sets and ` cliques. A graph is (r, `) if it admits an (r, `)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is (r, `)-well-covered if it is both (r, `) and well-covered. In this paper we consider two different decision problems. In the (r, `)-Well-Covered Graph problem ((r, `)wcg for short), we are given a graph G, and the question is whether G is an (r, `)-well-covered graph. In the Well-Covered (r, `)-Graph problem (wc(r, `)g for short), we are given an (r, `)-graph G together with an (r, `)-partition of V (G) into r independent sets and ` cliques, and the question is whether G is well-covered. We classify most of these problems into P, coNP-complete, NP-complete, NP-hard, or coNP-hard. Only the cases wc(r, 0)g for r ≥ 3 remain open. In addition, we consider the parameterized complexity of these problems for several choices of parameters, such as the size α of a maximum independent set of the input graph, its neighborhood diversity, or the number ` of cliques in an (r, `)-partition. In particular, we show that the parameterized problem of deciding whether a general graph is well-covered parameterized by α can be reduced to the wc(0, `)g problem parameterized by `, and we prove that this latter problem is in XP but does not admit polynomial kernels unless coNP ⊆ NP/poly.