Compatible Spanning Trees in Simple Drawings of $K_n$

TL;DR

This paper investigates the compatibility graph of plane spanning trees in simple drawings of the complete graph, proving connectivity and linear diameter bounds for various classes of drawings and specific tree types.

Contribution

It establishes the connectivity of the compatibility graph for certain classes of drawings and specific tree structures, with bounds on the graph's diameter.

Findings

Compatibility graph is connected for cylindrical, monotone, and strongly c-monotone drawings.

The subgraph of compatibility graph induced by stars and similar trees is also connected.

Diameter of the compatibility graph is at most linear in the number of vertices.

Abstract

For a simple drawing $D$ of the complete graph $K_n$, two (plane) subdrawings are compatible if their union is plane. Let $\mathcal{T}_D$ be the set of all plane spanning trees on $D$ and $\mathcal{F}(\mathcal{T}_D)$ be the compatibility graph that has a vertex for each element in $\mathcal{T}_D$ and two vertices are adjacent if and only if the corresponding trees are compatible. We show, on the one hand, that $\mathcal{F}(\mathcal{T}_D)$ is connected if $D$ is a cylindrical, monotone, or strongly c-monotone drawing. On the other hand, we show that the subgraph of $\mathcal{F}(\mathcal{T}_D)$ induced by stars, double stars, and twin stars is also connected. In all cases the diameter of the corresponding compatibility graph is at most linear in $n$.