Face-hitting Dominating Sets in Planar Graphs

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 α n 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 $3n/7$.