Total Domination In Generalized θ-Graphs And M×n Ladder Graphs

Let G=(V, E) be a graph without isolated vertices. A set S V (G) is a total dominatingset, denoted TDS, if every vertex in V is adjacent to a vertex in S. The total domination numberof G, denoted by γt(G), is the minimum cardinality of a total dominating set. A total dominatingset of G of cardinality γt(G) is called a γt(G)-set. G=(V, E) is called a generalized θ-graph if G is a simple connected graph obtained from two vertices x and y by adding at leasttwo paths joining x and y such that d(v)=2for each v∈V\{x, y}. For each m, n≥3andm, n∈Z, G=Lm×n=(V, E) is called a m×n ladder graph, if G is a simple connected graphobtained from two paths u1u2...umand v1v2...vmby adding a path of n1edges joining uiand vifor each i∈Im, such that d(v)=2for each v∈V\{ui, vi: i∈Im}. In this paper, thetotal domination numbers of generalized θ-graphs and m×n ladder graphs are determined.