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The Price of Connectivity for Vertex Cover: Perfect, Near-Perfect and Critical Graphs

In this paper we investigate the ratio of the connected vertex cover number to the vertex cover number in graphs, called the Price of Connectivity (PoC). For general graphs, this ratio is strictly bounded by 2. We prove that for every (P5, C5, C4)-free graph the ratio equals 1. We prove also that for every (P5, C4)-free graph the ratio is bounded by 4/3 and that for every (P7, C6,∆1,∆2)-free graph the ratio is bounded by 3/2, where∆1 and∆2 are two particular graphs. These results directly yields forbidden induced subgraphs characterizations of those graphs for which the PoC of every induced subgraph is bounded by t, for t ∈ {1, 4/3, 3/2}. Furthermore, we investigate PoC-critical graphs, namely, graphs whose PoC is strictly greater than the PoC of any proper induced subgraph.