WALCOM 2013: WALCOM: Algorithms and Computation pp 310-321

# On Graphs That Are Not PCGs

• Stephane Durocher
• Debajyoti Mondal
• Md. Saidur Rahman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7748)

## Abstract

Let T be an edge weighted tree and let d min ,d max be two nonnegative real numbers. Then the pairwise compatibility graph (PCG) of T 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 in T is in the interval [d min ,d max ]. Similarly, a given graph G is a PCG if there exist suitable T,d min ,d max , such that G is a PCG of T. Yanhaona, Bayzid and Rahman proved that there exists a graph with 15 vertices that is not a PCG. On the other hand, Calamoneri, Frascaria and Sinaimeri proved that every graph with at most seven vertices is a PCG. In this paper we construct a graph of eight vertices that is not a PCG, which strengthens the result of Yanhaona, Bayzid and Rahman, and implies optimality of the result of Calamoneri, Frascaria and Sinaimeri. We then construct a planar graph with sixteen vertices that is not a PCG. Finally, we prove a variant of the PCG recognition problem to be NP-complete.

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## Authors and Affiliations

• Stephane Durocher
• 1
• Debajyoti Mondal
• 1
• Md. Saidur Rahman
• 2
1. 1.Department of Computer ScienceUniversity of ManitobaCanada
2. 2.Department of Computer Science and Engineering, Graph Drawing & Information Visualization LaboratoryBangladesh University of Engineering and TechnologyBangladesh