A pair of trees without a simultaneous geometric embedding in the plane

TL;DR

This paper demonstrates that certain pairs of trees and a planar graph with a path cannot be simultaneously embedded in the plane without crossings, highlighting limitations in simultaneous geometric embeddings.

Contribution

The paper provides explicit examples of pairs of trees and a planar graph that cannot be simultaneously embedded with given vertex correspondences.

Findings

Certain pairs of trees lack a simultaneous geometric embedding.

A planar graph cannot be embedded simultaneously with a path under fixed vertex correspondence.

Examples highlight limitations of simultaneous embeddings in planar graphs.

Abstract

Any planar graph has a crossing-free straight-line drawing in the plane. A simultaneous geometric embedding of two n-vertex graphs is a straight-line drawing of both graphs on a common set of n points, such that the edges withing each individual graph do not cross. We consider simultaneous embeddings of two labeled trees, with predescribed vertex correspondences, and present an instance of such a pair that cannot be embedded. Further we provide an example of a planar graph that cannot be embedded together with a path when vertex correspondences are given.