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On Dilworth k Graphs and Their Pairwise Compatibility

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Algorithms and Computation (WALCOM 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8344))

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Abstract

The Dilworth number of a graph is the size of the largest subset of its nodes in which the close neighborhood of no node contains the neighborhood of another one. In this paper we give a new characterization of Dilworth k graphs, for each value of k, exactly defining their structure. Moreover, we put these graphs in relation with pairwise compatibility graphs (PCGs), i.e. graphs on n nodes that can be generated from an edge-weighted tree T that has n leaves, each representing a different node of the graph; two nodes are adjacent in the graph if and only if the weighted distance in the corresponding T is between two given non-negative real numbers, m and M. When either m or M are not used to eliminate edges from G, the two subclasses leaf power and minimum leaf power graphs (LPGs and mLPGs, respectively) are defined. Here we prove that graphs that are either LPGs or mLPGs of trees obtained connecting the centers of k stars with a path are Dilworth k graphs. We show that the opposite is true when k = 1,2, but not when k ≥ 3. Finally, we show that the relations we proved between Dilworth k graphs and chains of k stars hold only for LPGs and mLPGs, but not for PCGs.

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Authors and Affiliations

  1. Department of Computer Science, “Sapienza” University of Rome, Italy

    Tiziana Calamoneri & Rossella Petreschi

Authors
  1. Tiziana Calamoneri
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  2. Rossella Petreschi
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Editors and Affiliations

  1. Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, 721302, Kharagpur, India

    Sudebkumar Prasant Pal

  2. National Institute of Informatics, 2-1-2 Hitotsubashi, 101-8430, Chiyoda-ku, Tokyo, Japan

    Kunihiko Sadakane

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Calamoneri, T., Petreschi, R. (2014). On Dilworth k Graphs and Their Pairwise Compatibility. In: Pal, S.P., Sadakane, K. (eds) Algorithms and Computation. WALCOM 2014. Lecture Notes in Computer Science, vol 8344. Springer, Cham. https://doi.org/10.1007/978-3-319-04657-0_21

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  • DOI: https://doi.org/10.1007/978-3-319-04657-0_21

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-04656-3

  • Online ISBN: 978-3-319-04657-0

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