We initiate the study of Simultaneous Graph Embedding with Fixed Edges in the beyond planarity framework. In the QSEFE problem, we allow edge crossings, as long as each graph individually is drawn quasiplanar, that is, no three edges pairwise cross. %We call this problem the QSEFE problem. We show that a triple consisting of two planar graphs and a tree admit a QSEFE. This result also implies that a pair consisting of a 1-planar graph and a planar graph admits a QSEFE. 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 QSEFE. Moreover, in contrast to the planar SEFE problem, we show that it is not always possible to obtain a QSEFE for two matchings if the quasiplanar drawing of one matching is fixed.
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Packing Trees into 1-Planar Graphs
We introduce and study the 1-planar packing problem: Given k graphs with n vertices πΊ1,β¦,πΊπ, find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each πΊπ is a tree and π=3 . We prove that a triple consisting of three caterpillars or of two caterpillars and a path may not admit a 1-planar packing, while two paths and a special type of caterpillar always have one. We then study 1-planar packings with few crossings and prove that three paths (resp. cycles) admit a 1-planar packing with at most seven (resp. fourteen) crossings. We finally show that a quadruple consisting of three paths and a perfect matching with πβ₯12 vertices admits a 1-planar packing, while such a packing does not exist if πβ€10 .
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- Award ID(s):
- 1712119
- NSF-PAR ID:
- 10179543
- Date Published:
- Journal Name:
- 14th International Conference and Workshop on Algorithms and Computation (WALCOM)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-39881-1_8
