Collective additive tree 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, OH 44242, USA;Department of Computer Science, University of Toronto, Toronto, Ontario M5S 3G4, Canada;Mathematisches Institut, Brandenburgische Technische Universität Cottbus, D-03013 Cottbus, Germany;Department of Biomedical Informatics and OSUCCC Biomedical Informatics Shared Resource, Ohio State University, Columbus, OH 43210, USA

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2012

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Abstract

A graph G=(V,E) is said to admit a system of @m collective additive tree r-spanners if there is a system T(G) of at most @m spanning trees of G such that for any two vertices u,v of G a spanning tree T@?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 2log"3"2n collective additive tree 2-spanners and every n-vertex k-polygonal graph admits a system of at most 2log"3"2k+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 three 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.