Graph Sandwich Problem for the Property of Being Well-Covered and Partitionable into k Independent Sets and ℓ 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.