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 (2008) as the ratio between invariants from the connected version and the original version of the problem.
Camby, Cardinal, Fiorini and Schaudt (2014) for the vertex cover problem, Camby and Schaudt (2014) 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 et al. (2014) 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 (2016). Moreover, for the dominating set problem, we investigate critical trees and we show that every minimum dominating set of a critical graph is independent.
Published March 2017 , 16 pages