Domination Value in Graphs

Abstract

A set $D\subseteq V(G)$ is a \emph{dominating set} of $G$ if every vertex not in $D$ is adjacent to at least one vertex in $D$. A dominating set of $G$ of minimum cardinality is called a $\gamma (G)$-set. For each vertex $v\in V(G)$, we define the \emph{domination value} of $v$ to be the number of $\gamma (G)$-sets to which $v$ belongs. In this paper, we study some basic properties of the domination value function, thus initiating \emph{a local study of domination} in graphs. Further, we characterize domination value for the Petersen graph, complete $n$-partite graphs, cycles, and paths.

14 pages, 5 figures, v2: A few minor changes made. This is the final version, to appear in Contrib. Discrete Math

References in corpus (1)

• Total Domination Value in Graphs

Cited by in corpus (1)

• Domination Value in $P_2\square P_n$ and $P_2\square C_n$