Additive Spanners for Circle Graphs and Polygonal Graphs

  • Authors:

  • Feodor F. Dragan;Derek G. Corneil;Ekkehard Köhler;Yang Xiang

  • Affiliations:

  • Algorithmic Research Laboratory, Department of Computer Science, Kent State University, Kent, USA OH 44242;Department of Computer Science, University of Toronto, Toronto, Canada;Mathematisches Institut, Brandenburgische Technische Universität Cottbus, Cottbus, Germany D-03013;Algorithmic Research Laboratory, Department of Computer Science, Kent State University, Kent, USA OH 44242

  • Venue:
  • Graph-Theoretic Concepts in Computer Science
  • Year:
  • 2008

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Abstract

A graph G = (V ,E ) is said to admit a system of μ collective 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 u ,v of G a spanning tree $T\in \cal{T}$(G ) exists such that the distance in T between u and v is at most r plus their distance in G . In this paper, we examine the problem of finding "small" systems of collective additive tree r -spanners for small values of r on circle graphs and on polygonal graphs. Among other results, we show that every n -vertex circle graph admits a system of at most $2\log_{\frac{3}{2}}n$ collective additive tree 2-spanners and every n -vertex k -polygonal graph admits a system of at most $2\log_{\frac{3}{2}}k+7$ collective additive tree 2-spanners. Moreover, we show that every n -vertex k -polygonal graph admits an additive (k + 6)-spanner with at most 6n *** 6 edges and every n -vertex 3-polygonal graph admits a system of at most 3 collective additive tree 2-spanners and an additive tree 6-spanner. All our collective tree spanners as well as all sparse spanners are constructible in polynomial time.