Planar Embeddings with Small and Uniform Faces

  • Giordano Da Lozzo
  • Vít Jelínek
  • Jan Kratochvíl
  • Ignaz Rutter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8889)

Abstract

Motivated by finding planar embeddings that lead to drawings with favorable aesthetics, we study the problems MinMaxFace and UniformFaces of embedding a given biconnected multi-graph such that the largest face is as small as possible and such that all faces have the same size, respectively. We prove a complexity dichotomy for MinMaxFace and show that deciding whether the maximum is at most \(k\) is polynomial-time solvable for \(k \le 4\) and NP-complete for \(k \ge 5\). Further, we give a 6-approximation for minimizing the maximum face in a planar embedding. For UniformFaces, we show that the problem is NP-complete for odd \(k \ge 7\) and even \(k \ge 10\). Moreover, we characterize the biconnected planar multi-graphs admitting 3- and 4-uniform embeddings (in a \(k\)-uniform embedding all faces have size \(k\)) and give an efficient algorithm for testing the existence of a 6-uniform embedding.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Giordano Da Lozzo
    • 1
  • Vít Jelínek
    • 2
  • Jan Kratochvíl
    • 3
  • Ignaz Rutter
    • 3
    • 4
  1. 1.Department of EngineeringRoma Tre UniversityRomeItaly
  2. 2.Computer Science Institute, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  3. 3.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  4. 4.Faculty of InformaticsKarlsruhe Institute of Technology (KIT)KarlsruheGermany

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