Abstract If k ≥ 1 and G = ( V , E ) is a finite connected graph, S ⊆ V is said a distance k -dominating set if every vertex v ∈ V is within distance k from some vertex of S . The distance k -domination number γ w k ( G ) is the minimum cardinality among all distance k -dominating sets of G . A set S ⊆ V is a total dominating set if every vertex v ∈ V satisfies δ S ( v ) ≥ 1 and the total domination number, denoted by γ t ( G ) , is the minimum cardinality among all total dominating sets of G . The study of hyperbolic graphs is an interesting topic since the hyperbolicity of any geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this paper we obtain relationships between the hyperbolicity constant δ ( G ) and some domination parameters of a graph G . The results in this work are inequalities, such as γ w k ( G ) ≥ 2 δ ( G ) ∕ ( 2 k + 1 ) and δ ( G ) ≤ γ t ( G ) ∕ 2 + 3 .

中文翻译：

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双曲线图的支配

摘要 如果 k ≥ 1 且 G = ( V , E ) 是有限连通图，如果每个顶点 v ∈ V 都在距离 S 的某个顶点的距离 k 内，则称 S ⊆ V 是距离 k 支配集。距离 k 支配数 γ wk ( G ) 是 G 的所有距离 k 支配集中的最小基数。如果每个顶点 v ∈ V 满足 δ S ( v ) ≥ 1，则集合 S ⊆ V 是总支配集，并且总支配数用 γ t ( G ) 表示，是 G 的所有总支配集中的最小基数。双曲图的研究是一个有趣的话题，因为任何测地线度量空间的双曲性等价于与之相关的图的双曲性。在本文中，我们获得了双曲线常数 δ ( G ) 与图 G 的一些支配参数之间的关系。这项工作的结果是不平等，