New Results on Pairwise Compatibility Graphs

A graph $G=(V,E)$ is called a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$ such that each leaf $u$ of $T$ corresponds to a vertex $u \in V$ and there is an edge $(u, v) \in E$ if and only if $d_{min} \leq d_{T}(u, v) \leq d_{max}$, where $d_T(u, v)$ is the sum of the weights of the edges on the unique path from $u$ to $v$ in $T$. The tree $T$ is called the pairwise compatibility tree (PCT) of $G$. It has been proven that not all graphs are PCGs. Thus, it is interesting to know which classes of graphs are PCGs. In this paper, we prove that grid graphs are PCGs. Although there are a necessary condition and a sufficient condition known for a graph being a PCG, there are some classes of graphs that are intermediate to the classes defined by the necessary condition and the sufficient condition. In this paper, we show two examples of graphs that are included in these intermediate classes and prove that they are not PCGs.