TOTAL VERTEX COVER OF GRAPHS

Abstract

A vertex cover $S\subseteq V(G)$ is called a total vertex cover of $G$ if the graph $\langle S \rangle$ induced by set $S$ does not contain isolated vertices, i.e., $ | N_G(v) \cap S | \ge 1$ for every $v \in S$. The total vertex cover number of $G$, denoted by $\beta_t(G)$, is the minimum cardinality of a total vertex covering of $G$. In this paper, we show that given two positive integers $a$ and $b$ such that $2 \le a \le b$, there exist a connected graph $G$ such that $\gamma _t(G) = a$ and $\beta_t(G) = b$, where $\gamma _t(G)$ is the total domination number of $G$. We also characterize the total vertex covers of the join, corona, edge corona, and lexicographic product of two graphs. From these characterizations, we determine a bound or the exact value of the total vertex cover number of each of these graphs.