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 Citations (Scopus)

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 publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
PublisherSpringer Verlag
Pages1-13
Number of pages13
Volume9214
ISBN (Print)9783319218397
DOIs
StatePublished - 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)03029743
ISSN (Electronic)16113349

Other

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

Fingerprint

Sparse Graphs
Arc of a curve
Contact
Graph in graph theory
Maximum Degree
Interior
Representability
Plane Graph
Vertex of a graph
Line segment
Wedge
Convex Hull
Straight Line
Subgraph
Disjoint
NP-complete problem
Computing

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

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 Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9214, 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

Contact graphs of circular arcs. / Alam, Md Jawaherul; Eppstein, David; Kaufmann, Michael; Kobourov, Stephen G; Pupyrev, Sergey; Schulz, André; Ueckerdt, Torsten.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 9214 Springer Verlag, 2015. p. 1-13 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9214).

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

Alam, MJ, Eppstein, D, Kaufmann, M, Kobourov, SG, Pupyrev, S, Schulz, A & Ueckerdt, T 2015, Contact graphs of circular arcs. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 9214, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9214, Springer Verlag, pp. 1-13, 14th International Symposium on Algorithms and Data Structures, WADS 2015, Victoria, Canada, 8/5/15. https://doi.org/10.1007/978-3-319-21840-3_1
Alam MJ, Eppstein D, Kaufmann M, Kobourov SG, Pupyrev S, Schulz A et al. Contact graphs of circular arcs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 9214. Springer Verlag. 2015. p. 1-13. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-21840-3_1
Alam, Md Jawaherul ; Eppstein, David ; Kaufmann, Michael ; Kobourov, Stephen G ; Pupyrev, Sergey ; Schulz, André ; Ueckerdt, Torsten. / Contact graphs of circular arcs. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 9214 Springer Verlag, 2015. pp. 1-13 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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