Vol. 12, No. 5, 2019

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Pairwise compatibility graphs: complete characterization for wheels

Matthew Beaudouin-Lafon, Serena Chen, Nathaniel Karst, Denise Sakai Troxell and Xudong Zheng

Vol. 12 (2019), No. 5, 871–882

A simple graph G is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T with positive weights and nonnegative numbers dmin and dmax such that the leaves of T are exactly the vertices of G, and uv is an edge in G if and only if the sum of weights of edges on the unique path between u and v in T is at least dmin and at most dmax. We show that a wheel on n vertices is a PCG if and only if n 8, settling an open problem proposed by Calamoneri and Sinaimeri (SIAM Review 58:3 (2016), 445–460). Our approach is based on unavoidable binary classifications of the edges in the complement of wheels that are PCGs. (Note: during the review process of our work, we learned that the same result has been obtained independently with an alternative proof.)

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pairwise compatibility graph, PCG, phylogenetic tree, wheel
Mathematical Subject Classification 2010
Primary: 05C12, 05C78
Received: 27 September 2018
Revised: 28 January 2019
Accepted: 30 January 2019
Published: 22 May 2019

Communicated by Ann N. Trenk
Matthew Beaudouin-Lafon
Franklin W. Olin College of Engineering
Needham, MA
United States
Serena Chen
Franklin W. Olin College of Engineering
Needham, MA
United States
Nathaniel Karst
Mathematics and Sciences Division
Babson College
Babson Park, MA
United States
Denise Sakai Troxell
Mathematics and Sciences Division
Babson College
Babson Park, MA
United States
Xudong Zheng
Johns Hopkins University
Baltimore, MD
United States