On the (parameterized) complexity of recognizing well-covered (r, ℓ)-graph

Abstract An ( r , l ) -partition of a graph G is a partition of its vertex set into r independent sets and l cliques. A graph is ( r , l ) if it admits an ( r , l ) -partition. A graph is well-covered if every maximal independent set is also maximum. A graph is ( r , l ) -well-covered if it is both ( r , l ) and well-covered. In this paper we consider two different decision problems. In the ( r , l ) -Well-Covered Graph problem ( ( r , l ) wc-g for short), we are given a graph G, and the question is whether G is an ( r , l ) -well-covered graph. In the Well-Covered ( r , l ) -Graph problem ( wc- ( r , l ) g for short), we are given an ( r , l ) -graph G together with an ( r , l ) -partition, and the question is whether G is well-covered. This generates two infinite families of problems, for any fixed non-negative integers r and l, which we classify as being 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, its clique-width, or the number l of cliques in an ( r , l ) -partition. In particular, we show that the parameterized problem of determining whether every maximal independent set of an input graph G has cardinality equal to k can be reduced to the wc- ( 0 , l ) g problem parameterized by l. In addition, we prove that both problems are coW[2] -hard but can be solved in XP -time.