Multi-interval Pairwise Compatibility Graphs

(Extended Abstract)
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10185)


Let T be an edge weighted tree and let \(d_{min}\), \(d_{max}\) be two non-negative real numbers where \(d_{min}\le d_{max}\). A pairwise compatibility graph (PCG) of T for \(d_{min}\), \(d_{max}\) is a graph G such that each vertex of G corresponds to a distinct leaf of T and two vertices are adjacent in G if and only if the weighted distance between their corresponding leaves lies within the interval \([d_{min},d_{max}]\). A graph G is a PCG if there exist an edge weighted tree T and suitable \(d_{min}\), \(d_{max}\) such that G is a PCG of T. Knowing that all graphs are not PCGs, in this paper we introduce a variant of pairwise compatibility graphs which we call multi-interval PCGs. A graph G is a multi-interval PCG if there exist an edge weighted tree T and some mutually exclusive intervals of nonnegative real numbers such that there is an edge between two vertices in G if and only if the distance between their corresponding leaves in T lies within any such intervals. If the number of intervals is k, then we call the graph a k-interval PCG. We show that every graph is a k-interval PCG for some k. We also prove that wheel graphs and a restricted subclass of series-parallel graphs are 2-interval PCGs.


Pairwise compatibility graphs Phylogenetic trees Series-parallel graphs 



We thank Kazuo Iwama of Kyoto University who pointed out this variant of the problem when the second author discussed the PCG problem with him in 2014.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Graph Drawing and Information Visualization Laboratory, Department of Computer Science and EngineeringBangladesh University of Engineering and TechnologyDhakaBangladesh

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