Abstract We establish that any connected cubic graph of order n > 6 has a minimum vertex-edge dominating set of at most 10 n 31 vertices, thus affirmatively answering the open question posed by Klostermeyer et al. in Discussiones Mathematicae Graph Theory, https://doi.org/10.7151/dmgt.2175 . On the other hand, we present an infinite family of cubic graphs whose γ v e ratio is equal to 2 7 . Finally, we show that the problem of determining the minimum γ v e -dominating set is NP-hard even in cubic planar graphs.

中文翻译：

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三次图中的顶点-边支配

摘要 我们确定任何 n > 6 阶连通三次图都具有最多 10 n 31 个顶点的最小顶点边支配集，从而肯定地回答了 Klostermeyer 等人提出的开放性问题。在讨论数学图论，https://doi.org/10.7151/dmgt.2175。另一方面，我们提出了一个无限的三次图族，其 γ ve 比率等于 2 7 。最后，我们表明即使在三次平面图中，确定最小 γ ve 支配集的问题也是 NP 难的。