Total Global Domination in Permutation Graphs

Abstract

A total dominating set $D$ of a graph $G_\pi=(V_\pi,E_\pi)$ is a total global dominating set if $D$ is also a total dominating set of $\bar G_\pi$. The total global domination number $\gamma_{tg}(G_\pi)$ of $G_\pi$ is the minimum cardinality of a total global dominating set. In this paper, we permutation characterize total global dominating sets and bounds are obtained for $\gamma_{tg}(G_\pi)$. we exhibit inequalities involving variations on domination numbers and vertex covering number. Finally we found the total global domination number of a permutation and also derived the total global domination number of permutation graph through the permutation.