Abstract:A dominating set of a graph $G$ is a subset $D$ of vertices such that every vertex not in $D$ is adjacent to at least one vertex in $D$. A dominating set $D$ is paired if the subgraph induced by its vertices has a perfect matching, and semipaired if every vertex in $D$ is paired with exactly one other vertex in $D$ that is within distance 2 from it. The paired domination number, denoted by $\gamma_{pr}(G)$, is the minimum cardinality of a paired dominating set of $G$, and the semipaired domination number, denoted by $\gamma_{pr2}(G)$, is the minimum cardinality of a semipaired 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 $\gamma_{pr}(G) \le 2 \lfloor \frac{n}{4} \rfloor$ for any near-triangulation $G$ of order $n\ge 4$, and that with some exceptions, $\gamma_{pr2}(G) \le \lfloor \frac{2n}{5} \rfloor$ for any near-triangulation $G$ of order $n\ge 5$.
From: Mercè Mora [view email]
[v1]
Fri, 22 Jul 2022 08:01:32 UTC (136 KB)
[v2]
Wed, 2 Sep 2026 11:56:09 UTC (246 KB)
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