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Ortho-Polygon Visibility Representations of Embedded Graphs

  • Emilio Di Giacomo
  • Walter Didimo
  • William S. Evans
  • Giuseppe Liotta
  • Henk Meijer
  • Fabrizio MontecchianiEmail author
  • Stephen K. Wismath
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)

Abstract

An ortho-polygon visibility representation of an n-vertex embedded graph G (OPVR of G) is an embedding preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of G is the minimum k such that every polygon has at most k reflex corners. We present polynomial time algorithms that test whether G has an OPVR and, if so, compute one of minimum vertex complexity. We argue that the existence and the vertex complexity of an OPVR of G are related to its number of crossings per edge and to its connectivity. Namely, we prove that if G is 1-plane (i.e., it has at most one crossing per edge) an OPVR of G always exists while this may not be the case if two crossings per edge are allowed. Also, if G is a 3-connected 1-plane graph, we can compute in O(n) time an OPVR of G whose vertex complexity is bounded by a constant. However, if G is a 2-connected 1-plane graph, the vertex complexity of any OPVR of G may be \(\varOmega (n)\). In contrast, we describe a family of 2-connected 1-plane graphs for which an embedding that guarantees constant vertex complexity can be computed. Finally, we present the results of an experimental study on the vertex complexity of OPVRs of 1-plane graphs.

Keywords

Outer Face Embed Graph Horizontal Visibility Expansion Cycle Reflex Corner 
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 AG 2016

Authors and Affiliations

  • Emilio Di Giacomo
    • 1
  • Walter Didimo
    • 1
  • William S. Evans
    • 2
  • Giuseppe Liotta
    • 1
  • Henk Meijer
    • 3
  • Fabrizio Montecchiani
    • 1
    Email author
  • Stephen K. Wismath
    • 4
  1. 1.Università Degli Studi di PerugiaPerugiaItaly
  2. 2.University of British ColumbiaVancouverCanada
  3. 3.University College RooseveltMiddelburgThe Netherlands
  4. 4.University of LethbridgeLethbridgeCanada

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