Contact Graphs of Circular Arcs

  • Md. Jawaherul Alam
  • David Eppstein
  • Michael Kaufmann
  • Stephen G. Kobourov
  • Sergey Pupyrev
  • André Schulz
  • Torsten Ueckerdt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

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 \(k\ge 3\) every (2, k)-sparse graph has a CCA-representation. Hence the question of CCA-representability is open for (2, k)-sparse graphs with \(0\le k\le 2\). We partially answer this question by computing CCA-representations 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).

Keywords

Hull Haas Pebble Rote Cobos 

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Md. Jawaherul Alam
    • 1
  • David Eppstein
    • 2
  • Michael Kaufmann
    • 3
  • Stephen G. Kobourov
    • 1
  • Sergey Pupyrev
    • 1
    • 4
  • André Schulz
    • 5
  • Torsten Ueckerdt
    • 6
  1. 1.Department of Computer ScienceUniversity of ArizonaTucsonUSA
  2. 2.Computer Science DepartmentUniversity of CaliforniaIrvineUSA
  3. 3.Wilhelm-Schickard-Institut Für InformatikUniversität TübingenTübingenGermany
  4. 4.Institute of Mathematics and Computer ScienceUral Federal UniversityYekaterinburgRussia
  5. 5.Institut Math. Logik und GrundlagenforschungUniversität MünsterMünsterGermany
  6. 6.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

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