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2019 Pairwise compatibility graphs: complete characterization for wheels
Matthew Beaudouin-Lafon, Serena Chen, Nathaniel Karst, Denise Sakai Troxell, Xudong Zheng
Involve 12(5): 871-882 (2019). DOI: 10.2140/involve.2019.12.871

Abstract

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 n8, 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.)

Citation

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Matthew Beaudouin-Lafon. Serena Chen. Nathaniel Karst. Denise Sakai Troxell. Xudong Zheng. "Pairwise compatibility graphs: complete characterization for wheels." Involve 12 (5) 871 - 882, 2019. https://doi.org/10.2140/involve.2019.12.871

Information

Received: 27 September 2018; Revised: 28 January 2019; Accepted: 30 January 2019; Published: 2019
First available in Project Euclid: 29 May 2019

zbMATH: 07072557
MathSciNet: MR3954301
Digital Object Identifier: 10.2140/involve.2019.12.871

Subjects:
Primary: 05C12 , 05C78

Keywords: pairwise compatibility graph , PCG , phylogenetic tree , wheel

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.12 • No. 5 • 2019
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