Face-hitting Dominating Sets in Planar Graphs
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917604594024448 |
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| author | Francis, P. Illickan, Abraham M. Jose, Lijo M. Rajendraprasad, Deepak |
| author_facet | Francis, P. Illickan, Abraham M. Jose, Lijo M. Rajendraprasad, Deepak |
| contents | A dominating set of a graph $G$ is a subset $S$ of its vertices such that each vertex of $G$ not in $S$ has a neighbor in $S$. A face-hitting set of a plane graph $G$ is a set $T$ of vertices in $G$ such that every face of $G$ contains at least one vertex of $T$. We show that the vertex-set of every plane (multi-)graph without isolated vertices, self-loops or $2$-faces can be partitioned into two disjoint sets so that both the sets are dominating and face-hitting. We also show that all the three assumptions above are necessary for the conclusion.
As a corollary, we show that every $n$-vertex simple plane triangulation has a dominating set of size at most $(1 - α)n/2$, where $αn$ is the maximum size of an independent set in the triangulation. Matheson and Tarjan [European J. Combin., 1996] conjectured that every plane triangulation with a sufficiently large number of vertices $n$ has a dominating set of size at most $n / 4$. Currently, the best known general bound for this is by Christiansen, Rotenberg and Rutschmann [SODA, 2024] who showed that every plane triangulation on $n > 10$ vertices has a dominating set of size at most $2n/7$. Our corollary improves their bound for $n$-vertex plane triangulations which contain a maximal independent set of size either less than $2n/7$ or more than $3n/7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02808 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Face-hitting Dominating Sets in Planar Graphs Francis, P. Illickan, Abraham M. Jose, Lijo M. Rajendraprasad, Deepak Combinatorics Discrete Mathematics A dominating set of a graph $G$ is a subset $S$ of its vertices such that each vertex of $G$ not in $S$ has a neighbor in $S$. A face-hitting set of a plane graph $G$ is a set $T$ of vertices in $G$ such that every face of $G$ contains at least one vertex of $T$. We show that the vertex-set of every plane (multi-)graph without isolated vertices, self-loops or $2$-faces can be partitioned into two disjoint sets so that both the sets are dominating and face-hitting. We also show that all the three assumptions above are necessary for the conclusion. As a corollary, we show that every $n$-vertex simple plane triangulation has a dominating set of size at most $(1 - α)n/2$, where $αn$ is the maximum size of an independent set in the triangulation. Matheson and Tarjan [European J. Combin., 1996] conjectured that every plane triangulation with a sufficiently large number of vertices $n$ has a dominating set of size at most $n / 4$. Currently, the best known general bound for this is by Christiansen, Rotenberg and Rutschmann [SODA, 2024] who showed that every plane triangulation on $n > 10$ vertices has a dominating set of size at most $2n/7$. Our corollary improves their bound for $n$-vertex plane triangulations which contain a maximal independent set of size either less than $2n/7$ or more than $3n/7$. |
| title | Face-hitting Dominating Sets in Planar Graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2403.02808 |