Graphs that Are Not Pairwise Compatible: A New Proof Technique (Extended Abstract)

  • Pierluigi Baiocchi
  • Tiziana CalamoneriEmail author
  • Angelo Monti
  • Rossella Petreschi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10979)


A graph \(G=(V,E)\) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers \(d_{min}\) and \(d_{max}\), \(d_{min} \le d_{max}\), such that each node \(u \in V\) is uniquely associated to a leaf of T and there is an edge \((u,v) \in E\) if and only if \(d_{min} \le d_{T} (u, v) \le d_{max}\), where \(d_{T} (u, v)\) is the sum of the weights of the edges on the unique path \(P_{T}(u,v)\) from u to v in T. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. Despite numerous efforts, a complete characterization of the PCG class is not known yet. In this paper we propose a new proof technique that allows us to show that some interesting classes of graphs have empty intersection with PCG. We demonstrate our technique by showing many graph classes that do not lie in PCG. As a side effect, we show a not pairwise compatibility planar graph with 8 nodes (i.e. \(C^2_8\)), so improving the previously known result concerning the smallest planar graph known not to be PCG.


Phylogenetic tree reconstruction problem Pairwise Compatibility Graphs (PCGs) PCG recognition problem 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Pierluigi Baiocchi
    • 1
  • Tiziana Calamoneri
    • 1
    Email author
  • Angelo Monti
    • 1
  • Rossella Petreschi
    • 1
  1. 1.Computer Science Department“Sapienza” University of RomeRomeItaly

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