total domination in cubic knödel graphs

A subset d of vertices of a graph g is a dominating set if for each u∈v(g)∖d, u is adjacent to some vertex v∈d. the domination number, γ(g) of g, is the minimum cardinality of a dominating set of g. a set d⊆v(g) is a total dominating set if for each u∈v(g), u is adjacent to some vertex v∈d. the total domination number, γt(g) of g, is the minimum cardinality of a total dominating set of g. for an even integer n≥2 and 1≤δ≤⌊log2n⌋, a knodel graph wδ,n is a δ-regular bipartite graph of even order n, with vertices (i,j), for i=1,2 and 0≤j≤n2−1, where for every j, 0≤j≤n2−1, there is an edge between vertex (1,j) and every vertex (2,(j +2^k−1) mod n/2), for k=0,1,…,δ−1. in this paper, we determine the total domination number in 3-regular knodel graphs w_3,n.