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

Abstract

Let $$G=(V,E)$$G=(V,E) be a simple graph without isolated vertices. A set $$S$$S of vertices is a total dominating set of a graph $$G$$G if every vertex of $$G$$G is adjacent to some vertex in $$S$$S. A paired dominating set of $$G$$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 $$C_n\square C_m$$CnźCm for $$m=3,4$$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 $$C_m$$Cm bundles over a cycle $$C_n$$Cn for $$m=3,4$$m=3,4. Moreover, we give the exact value for the total domination number of Cartesian product $$C_n\square C_5$$CnźC5 and some upper bounds of $$C_m$$Cm bundles over a cycle $$C_n$$Cn where $$m\ge 5$$mź5.