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Generalized Powers of Graphs and Their Algorithmic Use

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Algorithm Theory – SWAT 2006 (SWAT 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4059))

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Abstract

Motivated by the frequency assignment problem in heterogeneous multihop radio networks, where different radio stations may have different transmission ranges, we introduce two new types of coloring of graphs, which generalize the well-known Distance-k-Coloring. Let G=(V,E) be a graph modeling a radio network, and assume that each vertex v of G has its own transmission radius r(v), a non-negative integer. We define r-coloring (r  +  -coloring) of G as an assignment Φ: V↦{0,1,2,...} of colors to vertices such that Φ(u)=Φ(v) implies d G (u,v)>r(v)+r(u) (d G (u,v)>r(v)+r(u)+1, respectively). The r-Coloring problem (the r  +  -Coloring problem) asks for a given graph G and a radius-function r: V↦N∪{0}, to find an r-coloring (an r  + -coloring, respectively) of G with minimum number of colors. Using a new notion of generalized powers of graphs, we investigate the complexity of the r-Coloring and r  + -Coloring problems on several families of graphs.

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Author information

Authors and Affiliations

  1. FB Informatik, Universität Rostock, Albert–Einstein–Str. 21, D–18051, Rostock, Germany

    Andreas Brandstädt

  2. Department of Computer Science, Kent State University, Kent, OH, 44242, USA

    Feodor F. Dragan, Yang Xiang & Chenyu Yan

Authors
  1. Andreas Brandstädt
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  2. Feodor F. Dragan
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  3. Yang Xiang
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  4. Chenyu Yan
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Editor information

Editors and Affiliations

  1. MADALGO, Department of Computer Science, University of Aarhus, Denmark

    Lars Arge

  2. Institute of Mathematics and Computer Science, University of Latvia, Raiņa bulvāris 29, Rīga, LV-1459, Latvia

    Rusins Freivalds

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© 2006 Springer-Verlag Berlin Heidelberg

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Brandstädt, A., Dragan, F.F., Xiang, Y., Yan, C. (2006). Generalized Powers of Graphs and Their Algorithmic Use. In: Arge, L., Freivalds, R. (eds) Algorithm Theory – SWAT 2006. SWAT 2006. Lecture Notes in Computer Science, vol 4059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11785293_39

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  • DOI: https://doi.org/10.1007/11785293_39

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35753-7

  • Online ISBN: 978-3-540-35755-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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