The Connected and Tree Domination Number of \(P(n,k)\) for \(k = 4,6,8\)

Abstract

Let \(\gamma_c(G)\)be the connected domination number of \(G\) and \(\gamma_t(G)\) be the tree domination number of \(G\). In this paper, we study the connected domination number and tree domination of \(P(n,k)\), and show that \(\gamma_{tr}(P(n, 4)) = \gamma_c(P(n, 4)) = n-1\) for \(n \geq 17\), \(\gamma_{tr}(P(n, 6)) = \gamma_c(P(n, 6)) = n-1\) for \(n \geq 25\), and \(\gamma_{tr}(P(n,8)) = \gamma_c(P(n,8)) = n-1\) for \(n \geq 33\).