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)


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).


Plane Graph Maximum Degree Outgoing Edge Sparse Graph Incoming Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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