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

Abstract

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.

This work was supported by FAPERJ, CNPq and CAPES Brazilian Research Agencies.