Abstract 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 ) ≤ ⌊ 2 n 5 ⌋ for any near-triangulation G of order n ≥ 5 , with two exceptions.

中文翻译：

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平面三角剖分中的总支配

摘要 图 G = ( V , E ) 的总支配集是 V 的子集 D，使得 V 中的每个顶点都与 D 中的至少一个顶点相邻。G 的总支配数，用γ t (G ) 表示，是G 的总支配集的最小基数。近三角剖分是一个双连通平面图，它允许一个平面嵌入，使得它的所有面都是三角形，可能除了外面。我们在本文中表明，对于任何 n ≥ 5 阶的近三角剖分 G，γ t ( G ) ≤ ⌊ 2 n 5 ⌋，但有两个例外。