Proximity drawings of outerplanar graphs (extended abstract)

  • William Lenhart
  • Giuseppe Liotta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1190)

Abstract

A proximity drawing of a graph is one in which pairs of adjacent vertices are drawn relatively close together according to some proximity measure while pairs of non-adjacent vertices are drawn relatively far apart. The fundamental question concerning proximity drawability is: Given a graph G and a definition of proximity, is it possible to construct a proximity drawing of G? We consider this question for outerplanar graphs with respect to an infinite family of proximity drawings called β-drawings. These drawings include as special cases the well-known Gabriel drawings (when β=1), and relative neighborhood drawings (when β=2). We first show that all biconnected outerplanar graphs are β-drawable for all values of β such that 1 ≤ β ≤ 2. As a side effect, this result settles in the affirmative a conjecture by Lubiw and Sleumer [15, 17], that any biconnected outerplanar graph admits a Gabriel drawing. We then show that there exist biconnected outerplanar graphs that do not admit any convex β-drawing for 1 ≤ β ≤ 2. We also provide upper bounds on the maximum number of biconnected components sharing the same cut-vertex in a β-drawable connected outerplanar graph. This last result is generalized to arbitrary connected planar graphs and is the first non-trivial characterization of connected β-drawable graphs. Finally, a weaker definition of proximity drawings is applied and we show that all connected outerplanar graphs are drawable under this definition.

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Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • William Lenhart
    • 1
  • Giuseppe Liotta
    • 2
  1. 1.Department of Computer ScienceWilliams CollegeWilliamstown
  2. 2.Department of Computer ScienceBrown UniversityProvidence

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