Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges

J. Joseph Fowler, Michael Jünger, Stephen Kobourov, Michael Schulz

Research output: Chapter in Book/Report/Conference proceedingConference contribution

9 Scopus citations

Abstract

A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with fixed edges (SEFE) has been open. We give a necessary and sufficient condition for whether a SEFE exists for pairs of graphs whose union is homeomorphic to K 5 or K 3,3. This allows us to characterize the class of planar graphs that always have a SEFE with any other planar graph. We also characterize the class of biconnected outerplanar graphs that always have a SEFE with any other outerplanar graph. In both cases, we provide efficient algorithms to compute a SEFE. Finally, we provide a linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has a SEFE.

Original languageEnglish (US)
Title of host publicationGraph-Theoretic Concepts in Computer Science - 34th International Workshop, WG 2008, Revised Papers
Pages146-158
Number of pages13
DOIs
StatePublished - Dec 1 2008
Event34th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2008 - Durham, United Kingdom
Duration: Jun 30 2008Jul 2 2008

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5344 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other34th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2008
CountryUnited Kingdom
CityDurham
Period6/30/087/2/08

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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    Fowler, J. J., Jünger, M., Kobourov, S., & Schulz, M. (2008). Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges. In Graph-Theoretic Concepts in Computer Science - 34th International Workshop, WG 2008, Revised Papers (pp. 146-158). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5344 LNCS). https://doi.org/10.1007/978-3-540-92248-3_14