Contact graphs of circular arcs

Md Jawaherul Alam, David Eppstein, Michael Kaufmann, Stephen G. Kobourov, Sergey Pupyrev, André Schulz, Torsten Ueckerdt

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

2 Scopus citations

Abstract

We study representations of graphs by contacts of circular arcs, CCA-representations for short, where the vertices are interior-disjoint circular arcs in the plane and each edge is realized by an endpoint of one arc touching the interior of another. A graph is (2, k)-sparse if every s-vertex subgraph has at most 2s − k edges, and (2, k)-tight if in addition it has exactly 2n−k edges, where n is the number of vertices. Every graph with a CCA-representation is planar and (2, 0)-sparse, and it follows from known results that for k ≥ 3 every (2, k)-sparse graph has a CCA-representation. Hence the question of CCA-representability is open for (2, k)-sparse graphs with 0 ≤ k ≤ 2. We partially answer this question by computing CCArepresentations for several subclasses of planar (2, 0)-sparse graphs. Next, we study CCA-representations in which each arc has an empty convex hull. We show that every plane graph of maximum degree 4 has such a representation, but that finding such a representation for a plane (2, 0)-tight graph with maximum degree 5 is NP-complete. Finally, we describe a simple algorithm for representing plane (2, 0)-sparse graphs with wedges, where each vertex is represented with a sequence of two circular arcs (straight-line segments).

Original languageEnglish (US)
Title of host publicationAlgorithms and Data Structures - 14th International Symposium, WADS 2015, Proceedings
EditorsFrank Dehne, Jorg-Rudiger Sack, Ulrike Stege
PublisherSpringer-Verlag
Pages1-13
Number of pages13
ISBN (Print)9783319218397
DOIs
StatePublished - Jan 1 2015
Event14th International Symposium on Algorithms and Data Structures, WADS 2015 - Victoria, Canada
Duration: Aug 5 2015Aug 7 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9214
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other14th International Symposium on Algorithms and Data Structures, WADS 2015
CountryCanada
CityVictoria
Period8/5/158/7/15

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Alam, M. J., Eppstein, D., Kaufmann, M., Kobourov, S. G., Pupyrev, S., Schulz, A., & Ueckerdt, T. (2015). Contact graphs of circular arcs. In F. Dehne, J-R. Sack, & U. Stege (Eds.), Algorithms and Data Structures - 14th International Symposium, WADS 2015, Proceedings (pp. 1-13). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9214). Springer-Verlag. https://doi.org/10.1007/978-3-319-21840-3_1