Abstract In this paper, we study a parameter that is a relaxation of an important domination parameter, namely the paired domination. A set D of vertices in G is a semipaired dominating set of G if it is a dominating set of G and can be partitioned into 2-element subsets such that the vertices in each 2-set are at most distance two apart. The semipaired domination number, γ pr2(G), is the minimum cardinality of a semipaired dominating set of G. For a graph G without isolated vertices, the domination number γ(G), the paired domination number γ pr (G) and the semitotal domination number γ t2(G) are related to the semipaired domination numbers by the following inequalities: γ(G) ≤ γ t2(G) ≤ γ pr2(G) ≤ γ pr (G) ≤ 2γ(G). It means that 1 ≤ γ pr2(G)/γ(G) ≤ 2. In this paper, we characterize those trees that attain the lower bound and the upper bound, respectively.

中文翻译：

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树中的支配与半成对支配

摘要 在本文中，我们研究了一个参数，它是一个重要支配参数的松弛参数，即配对支配。如果 G 中的顶点集 D 是 G 的支配集，并且可以将其划分为 2 元素子集，使得每个 2 集中的顶点至多相距 2，则 G 中的顶点集 D 是 G 的半成对支配集。半成对支配数γ pr2(G) 是G 的半成对支配集的最小基数。对于没有孤立顶点的图G，支配数γ(G)、成对支配数γ pr (G) 和半总支配数 γ t2(G) 通过以下不等式与半成对支配数相关： γ(G) ≤ γ t2(G) ≤ γ pr2(G) ≤ γ pr (G) ≤ 2γ(G)。这意味着 1 ≤ γ pr2(G)/γ(G) ≤ 2。在本文中，我们表征那些达到下界和上界的树，