Collective tree 1-spanners for interval graphs

  • Authors:

  • Derek G. Corneil;Feodor F. Dragan;Ekkehard Köhler;Chenyu Yan

  • Affiliations:

  • Department of Computer Science, University of Toronto, Toronto, Ontario, Canada;Department of Computer Science, Kent State University, Kent, Ohio;Institut für Mathematik, Technische Universität Berlin, Berlin, Germany;Department of Computer Science, Kent State University, Kent, Ohio

  • Venue:
  • WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
  • Year:
  • 2005

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Abstract

In this paper we study the existence of a small set $\mathcal{T}$ of spanning trees that collectively “1-span” an interval graph G. In particular, for any pair of vertices u,v we require a tree $T \in \mathcal{T}$such that the distance between u and v in T is at most one more than their distance in G. We show that: – there is no constant size set of collective tree 1-spanners for interval graphs (even unit interval graphs), – interval graph G has a set of collective tree 1-spanners of size O(log D), where D is the diameter of G, – interval graphs have a 1-spanner with fewer than 2n – 2 edges. Furthermore, at the end of the paper we state other results on collective tree c-spanners for c 1 and other more general graph classes.