Facial matchings in plane graphs

Abstract

Let $G$ be a plane graph. Two edges of $G$ are facially adjacent if they are consecutive on the facial walk of a face of $G$. A set of edges $M$ in $G$ is a (facial) matching if no two edges of $M$ are (facially) adjacent in $G$. Matchings in graphs are well studied, since finding large matchings in graphs has many applications.

In this note we investigate facial matchings and present tight estimations on sizes of maximal facial matchings, maximum facial matchings, and perfect facial matchings in general plane graphs, bridgeless plane graphs, and plane triangulations.