Shooting Stars in Simple Drawings of $K_{m,n}$

TL;DR

This paper proves that every simple drawing of a complete bipartite graph $K_{m,n}$ contains a plane spanning tree rooted at any vertex, answering an open question in graph drawing theory.

Contribution

It establishes that for any simple drawing of $K_{m,n}$, a plane spanning tree rooted at any vertex always exists, confirming a longstanding open problem.

Findings

Every simple drawing of $K_{m,n}$ contains a shooting star rooted at any vertex.

The existence of plane spanning trees is guaranteed in all simple drawings of $K_{m,n}$.

The result applies to all vertices in the drawing, ensuring a rooted plane spanning tree at each.

Abstract

Simple drawings are drawings of graphs in which two edges have at most one common point (either a common endpoint, or a proper crossing). It has been an open question whether every simple drawing of a complete bipartite graph $K_{m,n}$ contains a plane spanning tree as a subdrawing. We answer this question to the positive by showing that for every simple drawing of $K_{m,n}$ and for every vertex $v$ in that drawing, the drawing contains a shooting star rooted at $v$, that is, a plane spanning tree containing all edges incident to $v$.