Graph Sandwich Problem for the Property of Being Well-Covered and Partitionable into k Independent Sets and $$\ell $$ Cliques

A $$(k, \ell )$$ -partition of a graph G is a partition of its vertex set into k independent sets and $$\ell $$ cliques. A graph is $$(k, \ell )$$ if it admits a $$(k, \ell )$$ -partition. A graph is well-covered if every maximal independent set is also maximum. A graph is $$(k,\ell )$$ -well-covered if it is both $$(k,\ell )$$ and well-covered. In 2018, Alves et al. provided a complete mapping of the complexity of the $$(k,\ell )$$ -Well-Covered Graph problem, in which given a graph G, it is asked whether G is a $$(k,\ell )$$ -well-covered graph. Such a problem is polynomial-time solvable for the subclasses (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), and (2, 0), and NP-hard or coNP-hard, otherwise. In the Graph Sandwich Problem for Property $$\Pi $$ we are given a pair of graphs $$G^1=(V,E^1)$$ and $$G^2=(V,E^2)$$ with $$E^1\subseteq E^2$$ , and asked whether there is a graph $$G=(V,E)$$ with $$E^1\subseteq E\subseteq E^2$$ , such that G satisfies the property $$\Pi $$ . It is well-known that recognizing whether a graph G satisfies a property $$\Pi $$ is equivalent to the particular graph sandwich problem where $$E^1=E^2$$ . Therefore, in this paper we extend previous studies on the recognition of $$(k,\ell )$$ -well-covered graphs by presenting a complexity analysis of Graph Sandwich Problem for the property of being $$(k,\ell )$$ -well-covered. Focusing on the classes that are tractable for the problem of recognizing $$(k,\ell )$$ -well-covered graphs, we prove that Graph Sandwich for $$(k,\ell )$$ -well-covered is polynomial-time solvable when $$(k,\ell )=(0,1),(1,0),(1,1)$$ or (0, 2), and NP-complete if we consider the property of being (1, 2)-well-covered.