Extension and Its Price for the Connected Vertex Cover Problem

Abstract

We consider extension variants of Vertex Cover and Independent Set, following a line of research initiated in [9]. In particular, we study the Ext-CVC and the Ext-NSIS problems: given a graph G = (V, E) and a vertex set U ⊆ V , does there exist a minimal connected vertex cover (respectively, a maximal non-separating independent set) S, such that U ⊆ S (respectively, U ⊇ 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 Ext-CVC to Ext-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.

In memory of Jérôme Monnot

“…He was strongly interested in computational complexity of graph problems, and very talented at providing sharp limits between tractability and intractability, both from a parameterized viewpoint (see "On the complexity of solution extension of optimization problems" [2]) and from a standard viewpoint (see for instance "Complexity and algorithms for constant diameter augmentation problems" [12] or "On the complexity of independent dominating set with obligations in graphs" [13]). Structural, algorithmic and complexity aspects of graph problems are in the heart of several other articles in this special issue, such as "Blocking total dominating sets via edge contractions" [7], "Recoloring subgraphs of K 2n for sports scheduling" [17], or "Extension and its price for the connected vertex cover problem" [11].…”

p-Edge/Vertex-Connected Vertex Cover: Parameterized and Approximation Algorithms

“…Related Work on Connected Hitting Set problems. Ghadikolaei et al [10] studied a variant of Connected Vertex Cover problem where a minimal connected vertex cover has to contain a fixed set of vertices. Ramanujan [21] proved that for every α > 1, Connected Feedback Vertex Set has an α-approximate polynomial kernel.…”