Let T be an edge weighted tree and let dmin, dmax 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 [dmin, d max]. Similarly, a given graph G is a PCG if there exist suitable T, dmin, dmax, 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. © 2013 Springer-Verlag.
CITATION STYLE
Durocher, S., Mondal, D., & Rahman, M. S. (2013). On graphs that are not PCGs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7748 LNCS, pp. 310–321). https://doi.org/10.1007/978-3-642-36065-7_29
Mendeley helps you to discover research relevant for your work.