半总支配数为其阶数一半的图

Pub Date : 2024-09-13 DOI:10.1017/s0004972724000509

JIE CHEN, SHOU-JUN XU

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引用次数: 0

摘要

在无孤立图 G 中，如果顶点子集 S 是 G 的支配集，且 S 中的每个顶点与 S 中另一个顶点的距离都在 2 以内，则该顶点子集 S 是 G 的半总支配集。G 的半总支配数用 $\gamma _{t2}(G)$ 表示，是 G 中半总支配集的最小心性。Goddard、Henning 和 McPillan ['图中的半总支配数'，Utilitas Math.在本文中，我们将描述所有半总支配数为其阶数一半的图的特征。

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GRAPHS WITH SEMITOTAL DOMINATION NUMBER HALF THEIR ORDER

In an isolate-free graph G, a subset S of vertices is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number of G, denoted by $\gamma _{t2}(G)$, is the minimum cardinality of a semitotal dominating set in G. Goddard, Henning and McPillan [‘Semitotal domination in graphs’, Utilitas Math. 94 (2014), 67–81] characterised the trees and graphs of minimum degree 2 with semitotal domination number half their order. In this paper, we characterise all graphs whose semitotal domination number is half their order.

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