Total domination in cubic Knodel 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 somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set of G. For an eveninteger $nge 2$ and $1\le Delta \le lfloorlog_2nrfloor$, aKnodel graph $W_{Delta,n}$ is a $Delta$-regularbipartite graph of even order n, with vertices (i,j), for$i=1,2$ and $0le jle n/2-1$, where for every $j$, $0le jlen/2-1$, there is an edge between vertex $(1, j)$ and every vertex$(2,(j+2^k-1)$ mod (n/2)), for $k=0,1,cdots,Delta-1$. In thispaper, we determine the total domination number in $3$-regularKnodel graphs $W_{3,n}$.