Total perfect codes‎, ‎OO-irredundant and total subdivision in graphs

Document Type: Research Paper

Authors

Department of Mathematics‎, ‎Alzahra University‎, ‎P.O. Box 19834, Tehran‎, ‎Iran.

Abstract

‎Let $G=(V(G),E(G))$ be a graph‎, ‎$\gamma_t(G)$. Let $ooir(G)$ be the total domination and OO-irredundance number of $G$‎, ‎respectively‎. ‎A total dominating set $S$ of $G$ is called a $\textit{total perfect code}$ if every vertex in $V(G)$ is adjacent to exactly one vertex of $S$‎. ‎In this paper‎, ‎we show that if $G$ has a total perfect code‎, ‎then $\gamma_t(G)=ooir(G)$‎. ‎As a consequence, we determine the value of $ooir(G)$ for some classes of graphs‎.

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