The QuaSEFE Problem

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.

Efficiently Partitioning the Edges of a 1-Planar Graph into a Planar Graph and a Forest

“…Angelini et al [3] studied the problem of finding simultaneous quasi-planar drawings of graphs where some edges are fixed. They used Ackerman's partition result and cited its claimed linear runtime; with our result the linear runtime of [3] is established. It is known that every 1-planar graph has arboricity 4, i.e., its edges can be partitioned into 4 forests.…”