All Graphs with at most 8 nodes are 2-interval-PCGs

A graph G is a multi-interval PCG if there exist an edge weighted tree T with non-negative real values and disjoint intervals of the non-negative real half-line such that each node of G is uniquely associated to a leaf of T and there is an edge between two nodes in Gif and only if the weighted 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; in symbols, G= k-interval-PCG (T,I1,...,Ik). It is known that 2-interval-PCGs do not contain all graphs and the small- est known graph outside this class has 135 nodes. Here we prove that all graphs with at most 8 nodes are 2-interval-PCGs, so doing one step towards the determination of the smallest value of nsuch that there exists an nnode graph that is not a 2-interval-PCG.