Total domination in cubic Knödel graphs

Document Type : Original paper

Authors

1 Shahed University

2 Departtment of Mathematics, University of Mazandaran

3 Shahrood University of Technology

4 Tafresh University

Abstract

A subset D of vertices of a graph G is a dominating set if for each uV(G)D, u is adjacent to some vertex vD. The domination number, γ(G) of G, is the minimum cardinality of a dominating set of G. A set DV(G) is a total dominating set if for each uV(G), u is adjacent to some vertex vD. The total domination number, γt(G) of G, is the minimum cardinality of a total dominating set of G. For an even integer n2 and 1Δlog2n, a Kn\"odel graph WΔ,n is a Δ-regular bipartite graph of even order n, with vertices (i,j), for i=1,2 and 0jn21, where for every j, 0jn21, there is an edge between vertex (1,j) and every vertex (2,(j+2k1) mod n2), for k=0,1,,Δ1. In this paper, we determine the total domination number in 3-regular Kn\"odel graphs W3,n.

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