Pairwise Compatibility Graphs

Let T be an edge weighted tree, let dT (u, v) be the sum of the weights of the edges on the path from u to v in T , and let dmin and dmax be two non-negative real numbers such that dmin ≤ dmax. Then a pairwise compatibility graph of T for dmin and dmax is a graph G = (V, E), where each vertex u ∈ V corresponds to a leaf u of T and there is an edge (u, v) ∈ E if and only if dmin ≤ dT (u, v) ≤ dmax. A graph G is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers dmin and dmax such that G is a pairwise compatibility graph of T for dmin and dmax. In 2003, Kerney et al. introduced the concept of PCG and showed how to use them to model evolutionary relationship among a set of organisms. They conjectured that every graph is a PCG. In 2010, we have proved that there exists a graph of 15 vertices that is not a PCG. On the other hand, recently Calamoneri, Frascaria and Sinaimeri proved that every graph with at most seven vertices is a PCG. In this talk I show a graph of eight vertices that is not a PCG. Moreover, I show that several classes of graphs such as trees, cycles, cactus graphs, block graphs, ladder graphs are pairwise compatibility graphs. I also discuss the hardness of PCG recognition problem. The talk is based on my joint works with M. N. Yanhaona, M. S. Bayzid, K. S. M. T. Hossain, S. A. Salma, S. Durocher and D. Mondal.