## RESULTS ON RELATIVELY PRIME DOMINATION NUMBER OF VERTEX SWITCHING OF COMPLEMENT GRAPHS

## C. Jayasekaran and A. Jancy Vini

** DOI:**

https://doi.org/10.37418/amsj.9.4.15

Let $G$ be a non-trivial graph. A set $S\subseteq V$ is said to be a relatively prime dominating set if it is a dominating set with at least two elements and for every pair of vertices $u$ and $v$ in $S$ such that $(d(u), d(v))=1.$ The minimum cardinality of a relatively prime dominating set is called a relatively prime domination number and it is denoted by $\gamma_{\rm rpd}(G).$ For a finite undirected graph $G(V, E)$ and a subset $\sigma\subseteq V,$ the switching of $G$ by $\sigma$ is defined as the graph $G^{\sigma}(V,E’)$ which is obtained from $G$ by removing all edges between $\sigma$ and its complement $V-\sigma$ and adding as edges all non-edges between $\sigma$ and $V-\sigma.$ In this paper we compute the relatively prime domination number of vertex switching of complement of path $P_n,$ cycle $C_n,$ star $K_{1,n}$ and complete bipartite graph $K_{m,n}.$

**Keywords: **Dominating set, relatively prime dominating set, vertex switching.