Spanners in sparse graphs

  • Authors:

  • Feodor F. Dragan;Fedor V. Fomin;Petr A. Golovach

  • Affiliations:

  • Department of Computer Science, Kent State University, Kent, OH 44242, USA;Department of Informatics, University of Bergen, PB 7803, N-5020 Bergen, Norway;School of Engineering and Computing Sciences, Durham University, South Road, DH1 3LE Durham, UK

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2011

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Abstract

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t-spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs deciding whether G admits a tree t-spanner is polynomial time solvable for t==4. In this work we resolve the open question of Fekete and Kremer by proving much more general results:*The problem of finding a t-spanner of treewidth at most k in a given planar graph G is fixed parameter tractable parameterized by k and t. Moreover, for every fixed t and k, the running time of our algorithm is linear. *Our technique allows to extend the result from planar graphs to much more general classes of graphs. An apex graph is a graph that can be made planar by the removal of a single vertex. We prove that the problem of finding a t-spanner of treewidth k is fixed parameter tractable on graphs that do not contain some fixed apex graph as a minor, i.e. on apex-minor-free graphs. The class of apex-minor-free graphs contains planar graphs and graphs of bounded genus. *Finally, we show that the tractability border of the t-spanner problem cannot be extended beyond the class of apex-minor-free graphs and in this sense our results are tight. In particular, for every t=4, the problem of finding a tree t-spanner is NP-complete on K"6-minor-free graphs.