On relaxing the constraints in pairwise compatibility graphs

  • Authors:

  • Tiziana Calamoneri;Rossella Petreschi;Blerina Sinaimeri

  • Affiliations:

  • Department of Computer Science, "Sapienza" University of Rome, Roma, Italy;Department of Computer Science, "Sapienza" University of Rome, Roma, Italy;Department of Computer Science, "Sapienza" University of Rome, Roma, Italy

  • Venue:
  • WALCOM'12 Proceedings of the 6th international conference on Algorithms and computation
  • Year:
  • 2012

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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 dmin and dmax such that each leaf lu of T corresponds to a vertex u∈V and there is an edge (u,v)∈E if and only if dmin≤dT (lu, lv)≤dmax where dT (lu, lv) is the sum of the weights of the edges on the unique path from lu to lv in T. In this paper we analyze the class of PCG in relation with two particular subclasses resulting from the the cases where dmin=0 (LPG) and dmax=+∞ (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.