All Graphs with at most 8 nodes are 2-interval-PCGs
This result advances the characterization of 2-interval-PCGs, a problem in graph theory, by reducing the upper bound on the smallest counterexample.
The authors prove that all graphs with at most 8 nodes are 2-interval-PCGs, narrowing the search for the smallest graph outside this class from 135 nodes to at most 9 nodes.
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.