Partitions and well-coveredness: The graph sandwich problem

A graph G is well-covered if every maximal independent set of G is maximum. 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)-well-covered if it is well-covered and admits a (k, l)-partition. The recognition of (k, l)-well-covered graphs is polynomial-time solvable for the cases (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), and (2, 0), and hard, otherwise. In the graph sandwich problem for a property Pi, we are given a pair of graphs G(1) = (V, E-1) and G(2) = (V, E-2) with E-1 subset of E-2, and asked whether there is a graph G = (V, E) with E-1 subset of E subset of E-2, such that G satisfies the property Pi. The problem of recognizing whether a graph G satisfies a property Pi is equivalent to the particular graph sandwich problem where E-1 = E-2. In this paper, we study the graph sandwich problem for the property of being (k, l)-well-covered. We present some structural characterizations and extending previous studies on the recognition of (k, l)-well-covered graphs, we prove that GRAPH SANDWICH FOR (k, l)-WELL-COVEREDNESS is polynomial-time solvable when (k, l) is an element of{(0, 1), (1, 0), (1, 1), (0, 2)}. Besides, we show that it is NP-complete for the property of being (1, 2)-well-covered. (c) 2022 Elsevier B.V. All rights reserved.