Certified Vertex Cover of a Graph

Abstract

Let $G$ be a graph. Then $Q \subseteq V(G)$ is called a certified vertex covering set of G if $Q$ is a vertex cover of $G$ and every $x \in Q$, $x$ has zero or at least two neighbors in $V(G)\setminus Q$. The certified vertex cover number of $G$, denoted by $\beta_{cer}(G)$, is the minimum cardinality of a certified vertex cover of $G$. In this paper, we investigate this parameter on some special graphs and on the join of two graphs. We characterize certified vertex covering sets in these graphs and we use these results to derive the simplified formulas for solving the said parameter. Moreover, we present some bounds and properties of this parameter.