BOUNDS ON THE MULTIPLE DOMINATION NUMBER OF A SEMIGRAPH

S. Saravanan and P. Balaji

  DOI:  https://doi.org/10.37418/amsj.9.11.16

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The notion of $k-$domination in graphs was introduced by Fink and Jacobson\cite{fin}. S.S.Kamath and R.S.Bhat \cite{kam} introduced the concept of adjacency domination in semigraphs. They inspire us to define multiple domination number of semigraphs. Let $G=(V,X)$ be a semigraph and let $k$ be a positive integer. A set $D\subseteq V$ is called adjacent $k-$dominating set if every vertex $v\in V-D$ is adjacent to at least $k$ vertices of $D$. The adjacency $k-$domination number $\gamma_{k}^a$ is the minimum cardinality among the adjacent $k-$dominating sets of $G$. Also the end vertex adjacency $k-$domination number $\gamma_{k}^{ea}(G)$ is defined in the natural way. In this paper, the above multiple domination parameters are determined for various semigraphs, necessary and sufficient conditions and few bounds are obtained.

Keywords: Semigraph, adjacency $k-$domination number, end vertex adjacency $k-$domination number.