Extension and Its Price for the Connected Vertex Cover Problem.

We consider extension variants of VERTEX COVER and INDEPENDENT SET, following a line of research initiated in [10]. In particular, we study the Exr-CVC and the Exr-NSIS problems: given a graph G = (V, E) and a vertex set U subset of V, does there exist a minimal connected vertex cover (respectively, a maximal non-separating independent set) S, such that U subset of S (respectively, U superset of S). We present hardness results for both problems, for certain graph classes such as bipartite, chordal and weakly chordal. To this end we exploit the relation of Exr-CVC to Exr-VC, that is, to the extension variant of VERTEX COVER. We also study the Price of Extension (PoE), a measure that reflects the distance of a vertex set U to its maximum efficiently computable subset that is extensible to a minimal connected vertex cover, and provide negative and positive results for PoE in general and special graphs. (C) 2021 Published by Elsevier B.V.