Collective additive tree spanners of homogeneously orderable graphs

  • Authors:

  • Feodor F. Dragan;Chenyu Yan;Yang Xiang

  • Affiliations:

  • Algorithmic Research Laboratory, Department of Computer Science, Kent State University, Kent, OH;Algorithmic Research Laboratory, Department of Computer Science, Kent State University, Kent, OH;Algorithmic Research Laboratory, Department of Computer Science, Kent State University, Kent, OH

  • Venue:
  • LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
  • Year:
  • 2008

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Abstract

In this paper we investigate the (collective) tree spanners problem in homogeneously orderable graphs. This class of graphs was introduced by A. Brandstädt et al. to generalize the dually chordal graphs and the distance-hereditary graphs and to show that the Steiner tree problem can still be solved in polynomial time on this more general class of graphs. In this paper, we demonstrate that every n-vertex homogeneously orderable graph G admits - a spanning tree T such that, for any two vertices x, y of G, dT (x, y) ≤ dG(x, y) + 3 (i.e., an additive tree 3-spanner) and - a system T(G) of at most O(log n) spanning trees such that, for any two vertices x, y of G, a spanning tree T ∈ T(G) exists with dT (x, y) ≤ dG(x, y) + 2 (i.e, a system of at most O(log n) collective additive tree 2-spanners). These results generalize known results on tree spanners of dually chordal graphs and of distance-hereditary graphs. The results above are also complemented with some lower bounds which say that on some n-vertex homogeneously orderable graphs any system of collective additive tree 1-spanners must have at least Ω(n) spanning trees and there is no system of collective additive tree 2-spanners with constant number of trees.