Total Domination Value in Graphs

Details Total Domination Value in Graphs Total Domination Value in Graphs

2012-04-18

arxiv.org/abs/1204.3970v1

Mathematics

Combinatorics

17 pages, 6 figures, to appear in Util. Math. arXiv admin note: substantial text overlap with arXiv:1109.6277 by other author

Scientific paper

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.

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