On 2-Interval Pairwise Compatibility Properties of Two Classes of Grid Graphs

Abstract

Abstract A graph $G = (V,E)$ is called a pairwise compatibility graph (PCG) if it admits a tuple $(T, d_{min},d_{max})$ of an edge-weighted tree $T$ of non-negative edge weights with leaf set $L$, two non-negative real numbers $d_{min} \leq d_{max}$ such that each vertex $u^{\prime} \in V$ represents a leaf $u \in L$ and $G$ has an edge $(u^{\prime},v^{\prime}) \in E$ if and only if the distance between the two leaves $u$ and $v$ in the tree $T$ lies within interval $[d_{min}, d_{max}]$. It has been proven that not all graphs are PCGs. A graph $G$ is called a $k$-interval PCG if there exists an edge-weighted tree $T$ and $k$ mutually exclusive intervals of non-negative real numbers such that there is an edge between two vertices in $G$ if and only if the distance between their corresponding leaves in $T$ lies within any of the $k$ intervals. It is known that every graph $G$ is a $k$-interval PCG for $k=|E|$, where $E$ is the set of edges of $G$. It is thus interesting to know the smallest value of $k$ for which $G$ is a $k$-interval PCG. In this paper, we show that grid graphs and a subclass of $3$D grid graphs are $2$-interval PCGs.