All Graphs with at most 8 nodes are 2-interval-PCGs
Abstract
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 G if 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 smallest 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 n such that there exists an n node graph that is not a 2-interval-PCG.
Cite
@article{arxiv.2202.13844,
title = {All Graphs with at most 8 nodes are 2-interval-PCGs},
author = {Tiziana Calamoneri and Angelo Monti and Fabrizio Petroni},
journal= {arXiv preprint arXiv:2202.13844},
year = {2026}
}
Comments
7 pages, 3 figures, accepted to Fundamenta Informaticae