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

Abstract

A (k, ℓ) -partition of a graph G is a partition of its vertex set into k independent sets and ℓ cliques. A graph is (k, ℓ) if it admits a (k, ℓ) -partition. A graph is well-covered if every maximal independent set is also maximum. A graph is (k, ℓ) -well-covered if it is both (k, ℓ) and well-covered. In 2018, Alves et al. provided a complete mapping of the complexity of the (k, ℓ) -Well-Covered Graph problem, in which given a graph G, it is asked whether G is a (k, ℓ) -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 Π we are given a pair of graphs G1= (V, E1) and G2= (V, E2) with E1⊆ E2, and asked whether there is a graph G= (V, E) with E1⊆ E⊆ E2, such that G satisfies the property Π. It is well-known that recognizing whether a graph G satisfies a property Π is equivalent to the particular graph sandwich problem where E1= E2. Therefore, in this paper we extend previous studies on the recognition of (k, ℓ) -well-covered graphs by presenting a complexity analysis of Graph Sandwich Problem for the property of being (k, ℓ) -well-covered. Focusing on the classes that are tractable for the problem of recognizing (k, ℓ) -well-covered graphs, we prove that Graph Sandwich for (k, ℓ) -well-covered is polynomial-time solvable when (k, ℓ) = (0, 1 ), (1, 0 ), (1, 1 ) or (0, 2), and NP-complete if we consider the property of being (1, 2)-well-covered.