Core Concepts
The authors demonstrate the existence of dominating and face-hitting sets in plane graphs, highlighting the necessity of specific conditions for their conclusions.
Abstract
The content explores the concept of dominating sets and face-hitting sets in planar graphs. It discusses the partitioning of vertex sets into two disjoint subsets that are both dominating and face-hitting. The analysis includes corollaries related to simple plane triangulations and conjectures by Matheson and Tarjan. The paper also delves into polychromatic colorings, proving theorems, lemmas, and observations to support the main arguments. Furthermore, it provides applications to Matheson-Tarjan Conjecture and references previous works on domination numbers in planar graphs.
Stats
Every n-vertex simple plane triangulation has a dominating set of size at most (1 − α)n/2.
Currently, every plane triangulation on n > 10 vertices has a dominating set of size at most 2n/7.
For triangulated discs, the upper bound of n/3 by Matheson and Tarjan is tighter than general bounds.
Fomin and Thilikos showed that the k-dominating set problem on planar graphs can be solved in time O(215.13√k +n3).
Botler et al. proved that every planar triangulation G on n vertices has an independent dominating set of size less than 3n/8.
Quotes
"Every plane graph without isolated vertices, self-loops or 2-faces has a 2-coloring which is simultaneously domatic and polychromatic." - Theorem 1
"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." - Content Summary