Price of Connectivity for the vertex cover problem and the dominating set problem: conjectures and investigation of critical graphs

The vertex cover problem and the dominating set problem are two well-known problems in graph theory. Their goal is to find the minimum size of a vertex subset satisfying some properties. Both hold a connected version, which imposes that the vertex subset must induce a connected component. To study the interdependence between the connected version and the original version of a problem, the Price of Connectivity (PoC) was introduced by Cardinal and Levy [8, 14] as the ratio between invariants from the connected version and the original version of the problem. Camby, Cardinal, Fiorini and Schaudt [5] for the vertex cover problem, Camby and Schaudt [7] for the dominating set problem characterized some classes of PoC-Near-Perfect graphs, hereditary classes of graphs in which the Price of Connectivity is bounded by a fixed constant. Moreover, only for the vertex cover problem, Camby & al. [5] introduced the notion of critical graphs, graphs that can appear in the list of forbidden induced subgraphs characterization. By definition, the Price of Connectivity of a critical graph is strictly greater than that of any proper induced subgraph. In this paper, we prove that for the vertex cover problem, every critical graph is either isomorphic to a cycle on 5 vertices or bipartite. To go further in the previous studies, we also present conjectures on PoC-Near-Perfect graphs and critical graphs with the help of the computer software GraphsInGraphs [4]. Moreover, for the dominating set problem, we investigate critical trees and we show that every minimum dominating set of a critical graph is independent. keywords: vertex cover, connected vertex cover, dominating set, connected dominating set, forbidden induced subgraph, extremal graph.