Collective tree spanners and routing in AT-free related graphs

  • Authors:

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

  • Affiliations:

  • Department of Computer Science, Kent State University, Kent, Ohio;Department of Computer Science, Kent State University, Kent, Ohio;Department of Computer Science, University of Toronto, Toronto, Ontario, Canada

  • Venue:
  • WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
  • Year:
  • 2004

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Abstract

In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in AT-free graphs. We say that a graph G=(V,E) admits a system of μcollective additive tree r-spanners if there is a system ${\cal T}(G)$ of at most μ spanning trees of G such that for any two vertices x,y of G a spanning tree $T\in {\cal T}(G)$ exists such that dT(x,y)≤ dG(x,y)+r. Among other results, we show that AT-free graphs have a system of two collective additive tree 2-spanners (whereas there are trapezoid graphs that do not admit any additive tree 2-spanner). Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those graphs. Also, any DSP-graph (there exists a dominating shortest path) admits one additive tree 4-spanner, a system of two collective additive tree 3-spanners and a system of five collective additive tree 2-spanners.