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On Relaxing the Constraints in Pairwise Compatibility Graphs

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7157)

Abstract

A graph G is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers d min and d max such that each leaf l u of T corresponds to a vertex u ∈ V and there is an edge (u,v) ∈ E if and only if d min  ≤ d T (l u , l v ) ≤ d max where d T (l u , l v ) is the sum of the weights of the edges on the unique path from l u to l v in T. In this paper we analyze the class of PCG in relation with two particular subclasses resulting from the the cases where d min  = 0 (LPG) and d max  = + ∞ (mLPG). In particular, we show that the union of LPG and mLPG does not coincide with the whole class PCG, their intersection is not empty, and that neither of the classes LPG and mLPG is contained in the other. Finally, as the graphs we deal with belong to the more general class of split matrogenic graphs, we focus on this class of graphs for which we try to establish the membership to the PCG class.

Keywords

  • PCG
  • leaf power graph
  • threshold graphs
  • matrogenic graphs

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© 2012 Springer-Verlag Berlin Heidelberg

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Calamoneri, T., Petreschi, R., Sinaimeri, B. (2012). On Relaxing the Constraints in Pairwise Compatibility Graphs. In: Rahman, M.S., Nakano, Si. (eds) WALCOM: Algorithms and Computation. WALCOM 2012. Lecture Notes in Computer Science, vol 7157. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28076-4_14

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  • DOI: https://doi.org/10.1007/978-3-642-28076-4_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-28075-7

  • Online ISBN: 978-3-642-28076-4

  • eBook Packages: Computer ScienceComputer Science (R0)