Disjoint Compatibility via Graph Classes

TL;DR

This paper investigates the structure and connectivity of a graph formed by plane perfect matchings on convex point sets, focusing on compatibility via specific classes of plane graphs like spanning trees, caterpillars, and paths.

Contribution

It introduces a new graph model based on disjoint compatibility of plane perfect matchings and analyzes its connectivity and diameter for different classes of plane graphs.

Findings

The graph is connected with constant diameter when using all plane spanning trees.

The graph has linear diameter when using caterpillars.

The graph is disconnected when using paths.

Abstract

Two plane drawings of graphs on the same set of points are called disjoint compatible if their union is plane and they do not have an edge in common. Let $S$ be a convex point set of $2n \geq 10$ points and let $\mathcal{H}$ be a family of plane drawings on $S$. Two plane perfect matchings $M_1$ and $M_2$ on $S$ (which do not need to be disjoint nor compatible) are \emph{disjoint $\mathcal{H}$-compatible} if there exists a drawing in $\mathcal{H}$ which is disjoint compatible to both $M_1$ and $M_2$ In this work, we consider the graph which has all plane perfect matchings as vertices and where two vertices are connected by an edge if the matchings are disjoint $\mathcal{H}$-compatible. We study the diameter of this graph when $\mathcal{H}$ is the family of all plane spanning trees, caterpillars or paths. We show that in the first two cases the graph is connected with constant and linear diameter, respectively, while in the third case it is disconnected.