Total Domination Value in Graphs

Abstract

A set $D \subseteq V(G)$ is a \emph{total dominating set} of $G$ if for every vertex $v \in V(G)$ there exists a vertex $u \in D$ such that $u$ and $v$ are adjacent. A total dominating set of $G$ of minimum cardinality is called a $\gamma_t(G)$-set. For each vertex $v \in V(G)$, we define the \emph{total domination value} of $v$, $TDV(v)$, to be the number of $\gamma_t(G)$-sets to which $v$belongs. This definition gives rise to \emph{a local study of total domination} in graphs. In this paper, we study some basic properties of the $TDV$ function; also, we derive explicit formulas for the $TDV$ of any complete n-partite graph, any cycle, and any path.