On the (Parameterized) Complexity of Recognizing Well-Covered $(r,\ell )$-graphs

Abstract

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

This work was supported by FAPERJ, CNPq, CAPES Brazilian Research Agencies and EPSRC (EP/K025090/1).