Abstract
An efficient total dominating set D of a graph G is a vertex subset such that every vertex of G has exactly one neighbor in the set D. In this paper, we give necessary and sufficient conditions for the existence of efficient total domination sets of circulant graphs whose degree is 5 and classify these sets.
Highlights
All graphs considered in this paper are finite, simple, undirected, and connected
A total domination set of G with no isolated vertex is a subset D of V ( G ) such that every vertex in G is adjacent to a vertex in D
Please note that for an efficient total dominating set D of G, if u is an element of D, N2 (u) ∩ D = ∅
Summary
All graphs considered in this paper are finite, simple, undirected, and connected. For a graph G, we denote the vertex set and edge set by V ( G ) and E( G ), respectively. The Cayley graph C(Γ, X ) has vertex set Γ with any two vertices g, h ∈ Γ joined by an edge whenever g−1 h ∈ X. It follows that the Cayley graphs considered in this paper are finite, connected and undirected. Warren showed that the same methods can be used to relate efficient total dominating sets in Cayley graphs to covers of a reflexive complete graph [9].
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