Total and paired domination numbers of $C_m$ bundles over a cycle $C_n$

Abstract

Let $G=(V,E)$ be a simple graph without isolated vertices. A set $S$ of vertices is a total dominating set of a graph $G$ if every vertex of $G$ is adjacent to some vertex in $S$. A paired dominating set of $G$ is a dominating set whose induced subgraph has a perfect matching. The minimum cardinality of a total dominating set (respectively, a paired dominating set) is the total domination number (respectively, the paired domination number). Hu and Xu (J Combin Optim 27(2):369–378, 2014) computed the exact values of total and paired domination numbers of Cartesian product $C_n\square C_m$ for $m=3,4$. Graph bundles generalize the notions of covering graphs and Cartesian products. In this paper, we generalize these results given in Hu and Xu (J Combin Optim 27(2):369–378, 2014) to graph bundle and compute the total domination number and the paired domination number of $C_m$ bundles over a cycle $C_n$ for $m=3,4$. Moreover, we give the exact value for the total domination number of Cartesian product $C_n\square C_5$ and some upper bounds of $C_m$ bundles over a cycle $C_n$ where $m\ge 5$.