The QuaSEFE Problem

TL;DR

This paper explores the new QuaSEFE problem, allowing edge crossings under quasiplanar constraints, and establishes conditions for when simultaneous embeddings are possible for various graph classes.

Contribution

It introduces the QuaSEFE problem, providing new positive and negative results for simultaneous embeddings of quasiplanar and planar graphs.

Findings

A triple of two planar graphs and a tree admits a QuaSEFE.

A pair of a 1-planar graph and a planar graph admits a QuaSEFE.

Certain triples of planar graphs with specific subgraph properties also admit QuaSEFE.

Abstract

We initiate the study of Simultaneous Graph Embedding with Fixed Edges in the beyond planarity framework. In the QuaSEFE problem, we allow edge crossings, as long as each graph individually is drawn quasiplanar, that is, no three edges pairwise cross. We show that a triple consisting of two planar graphs and a tree admit a QuaSEFE. This result also implies that a pair consisting of a 1-planar graph and a planar graph admits a QuaSEFE. We show several other positive results for triples of planar graphs, in which certain structural properties for their common subgraphs are fulfilled. For the case in which simplicity is also required, we give a triple consisting of two quasiplanar graphs and a star that does not admit a QuaSEFE. Moreover, in contrast to the planar SEFE problem, we show that it is not always possible to obtain a QuaSEFE for two matchings if the quasiplanar drawing of one matching is fixed.