Certified Vertex Cover of a Graph (European Journal of Pure and Applied Mathematics, 2024)
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 $β_{cer}(G)$, 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.
Keywords: certified set; vertex cover; certified vertex covering set; certified vertex cover number