Face-hitting Dominating Sets in Planar Graphs
P. Francis, Abraham M. Illickan, Lijo M. Jose, Deepak Rajendraprasad
TL;DR
The paper proves that every plane graph without isolated vertices, self-loops, or $2$-faces admits a $2$-coloring that is simultaneously domatic and polychromatic. It achieves this via a dummy-edge augmentation that yields a supergraph $G'$ in which every $3^+$-face is made happy, followed by a $4$-coloring of $G'$ and a color-merge to obtain the desired two-coloring, with an extension argument to general graphs. A corollary shows that for $n$-vertex simple plane triangulations with a maximal independent set of size $\alpha n$, the domination number satisfies $\gamma(G) \le (1-\alpha)n/2$, which improves the bound $2n/7$ from Christiansen et al. for triangulations when $\alpha < 2/7$ or $\alpha > 3/7$, and thus advances Matheson-Tarjan-type bounds in those cases. The results connect domatic and polychromatic colorings in planar graphs and have implications for terrain guarding and domination in triangulations.
Abstract
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 $3n/7$.