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

TL;DR

This paper proves that all graphs with up to 8 nodes can be represented as 2-interval-PCGs, advancing understanding of the class's limitations.

Contribution

It establishes that every graph with at most 8 nodes is a 2-interval-PCG, narrowing the gap in characterizing these graph classes.

Findings

All graphs with ≤8 nodes are 2-interval-PCGs.

Smallest known non-2-interval-PCG has 135 nodes.

Progress towards identifying the minimal non-2-interval-PCG.

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.