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 $n\ge 17$, ${\gamma }_{tr}(P(n,6))={\gamma }_c(P(n,6))=n-1$ for $n\ge 25$, and ${\gamma }_{tr}(P(n,8))={\gamma }_c(P(n,8))=n-1$ for $n\ge 33$.