Abstract:In a recent paper, Francis, Illickan, Jose and Rajendraprasad showed that every $n$-vertex plane graph $G$ has (under some natural restrictions) a vertex-partition into two sets $V_1$ and $V_2$ such that each $V_i$ is \emph{dominating} (every vertex of $G$ contains a vertex of $V_i$ in its closed neighbourhood) and \emph{face-hitting} (every face of $G$ is incident to a vertex of $V_i$). Their proof works by considering a supergraph $G'$ of $G$ that has certain properties, and among all such graphs, taking one that has the fewest edges. As such, their proof is not algorithmic. Their proof also relies on the 4-color theorem, for which a quadratic-time algorithm exists, but it would not be easy to implement.
In this paper, we give a new proof that every $n$-vertex plane graph $G$ has (under the same restrictions) a vertex-partition into two dominating face-hitting sets. Our proof is constructive, and requires nothing more complicated than splitting a graph into 2-connected components, finding an ear decomposition, and computing a perfect matching in a 3-regular plane graph. For all these problems, linear-time algorithms are known and so we can find the vertex-partition in linear time.
From: Therese Biedl [view email]
[v1]
Fri, 15 Aug 2025 12:49:38 UTC (93 KB)
[v2]
Sat, 14 Mar 2026 22:00:15 UTC (330 KB)
[v3]
Wed, 17 Jun 2026 16:34:06 UTC (368 KB)
[v4]
Thu, 16 Jul 2026 20:58:07 UTC (368 KB)
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