On Simultaneous Graph Embedding

TL;DR

This paper investigates the problem of simultaneously embedding multiple planar graphs, providing algorithms and bounds for different variants with and without vertex mappings, and establishing impossibility results for certain cases.

Contribution

It introduces new algorithms and bounds for simultaneous graph embedding with and without vertex mappings, including positive results for certain graph classes and impossibility proofs for others.

Findings

Outerplanar graphs can be embedded on an O(n) x O(n) grid without mapping.

Embedding an outerplanar and a general planar graph requires an O(n^2) x O(n^3) grid.

Certain combinations of paths and caterpillars cannot be embedded simultaneously with given mappings.

Abstract

We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another where the mapping is not given. In particular, we show that without mapping, any number of outerplanar graphs can be embedded simultaneously on an $O(n)\times O(n)$ grid, and an outerplanar and general planar graph can be embedded simultaneously on an $O(n^2)\times O(n^3)$ grid. If the mapping is given, we show how to embed two paths on an $n \times n$ grid, a caterpillar and a path on an $n \times 2n$ grid, or two caterpillar graphs on an $O(n^2)\times O(n^3)$ grid. We also show that 5 paths, or 3 caterpillars, or two general planar graphs cannot be simultaneously embedded given the mapping.