Self-approaching Graphs

  • Soroush Alamdari
  • Timothy M. Chan
  • Elyot Grant
  • Anna Lubiw
  • Vinayak Pathak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

Abstract

In this paper we introduce self-approaching graph drawings. A straight-line drawing of a graph is self-approaching if, for any origin vertex s and any destination vertex t, there is an st-path in the graph such that, for any point q on the path, as a point p moves continuously along the path from the origin to q, the Euclidean distance from p to q is always decreasing. This is a more stringent condition than a greedy drawing (where only the distance between vertices on the path and the destination vertex must decrease), and guarantees that the drawing is a 5.33-spanner.

We study three topics: (1) recognizing self-approaching drawings; (2) constructing self-approaching drawings of a given graph; (3)constructing a self-approaching Steiner network connecting a given set of points.

We show that: (1) there are efficient algorithms to test if a polygonal path is self-approaching in ℝ2 and ℝ3, but it is NP-hard to test if a given graph drawing in ℝ3 has a self-approaching uv-path; (2) we can characterize the trees that have self-approaching drawings; (3) for any given set of terminal points in the plane, we can find a linear sized network that has a self-approaching path between any ordered pair of terminals.

Keywords

self-approaching increasing-chord graph drawing 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Soroush Alamdari
    • 1
  • Timothy M. Chan
    • 1
  • Elyot Grant
    • 2
  • Anna Lubiw
    • 1
  • Vinayak Pathak
    • 1
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

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