Multi-interval Pairwise Compatibility Graphs

Abstract

Let T be an edge weighted tree and let $d_{min}$, $d_{max}$ be two non-negative real numbers where $d_{min}\le d_{max}$. A pairwise compatibility graph (PCG) of T for $d_{min}$, $d_{max}$ 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 lies within the interval $[d_{min},d_{max}]$. A graph G is a PCG if there exist an edge weighted tree T and suitable $d_{min}$, $d_{max}$ such that G is a PCG of T. Knowing that all graphs are not PCGs, in this paper we introduce a variant of pairwise compatibility graphs which we call multi-interval PCGs. A graph G is a multi-interval PCG if there exist an edge weighted tree T and some mutually exclusive intervals of nonnegative 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 such intervals. If the number of intervals is k, then we call the graph a k-interval PCG. We show that every graph is a k-interval PCG for some k. We also prove that wheel graphs and a restricted subclass of series-parallel graphs are 2-interval PCGs.