Total domination in plane triangulations
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A total dominating set of a graph G=(V,E) is a subset D of V such that every vertex in V is adjacent to at least one vertex in D. The total domination number of G, denoted by γ _t (G), is the minimum cardinality of a total dominating set of G. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show in this paper that γ _t (G) ≤⌊2n/5⌋ for any near-triangulation G of order n≥ 5, with two exceptions.
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