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

Tiziana Calamoneri, Angelo Monti, Fabrizio Petroni

TL;DR

The paper establishes that every graph with at most $8$ nodes is a $2$-interval-PCG, advancing the understanding of the boundary between PCGs and multi-interval PCGs. It introduces structural results for universal and almost universal nodes to lift $k$-interval-PCG representations and applies an ILP-based approach restricted to full binary trees and caterpillars to certify the small hard cases. The authors certify that the eight-node graphs $G_2$, $G_3$, $G_5$, $G_4$, $G_6$, and $G_7$ are all $2$-interval-PCGs and provide explicit witness trees and interval pairs. The discussion acknowledges the complexity of nine-node graphs and outlines directions for extending the method to higher interval counts or larger graphs.

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.

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

TL;DR

The paper establishes that every graph with at most nodes is a -interval-PCG, advancing the understanding of the boundary between PCGs and multi-interval PCGs. It introduces structural results for universal and almost universal nodes to lift -interval-PCG representations and applies an ILP-based approach restricted to full binary trees and caterpillars to certify the small hard cases. The authors certify that the eight-node graphs , , , , , and are all -interval-PCGs and provide explicit witness trees and interval pairs. The discussion acknowledges the complexity of nine-node graphs and outlines directions for extending the method to higher interval counts or larger graphs.

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.
Paper Structure (4 sections, 7 theorems, 12 equations, 3 figures)

This paper contains 4 sections, 7 theorems, 12 equations, 3 figures.

Table of Contents

  1. Introduction
  2. $G_4$, $G_6$ and $G_7$ are $2$-interval-PCGs.
  3. $G_2$, $G_3$ and $G_5$ are $2$-interval-PCGs
  4. Conclusions

Key Result

Theorem 2.1

Let $G$ be a graph with a universal node $u$. If the subgraph $G'$ of $G$, obtained from $G$ by removing $u$, is a $k$-interval-PCG, then $G$ is a $(k+1)$-interval-PCG.

Figures (3)

  • Figure 1: The seven graphs with 8 nodes that are not PCGs.
  • Figure 2: The seven witness complete binary trees with 8 leaves and the corresponding intervals.
  • Figure 3: The seven witness caterpillars with 8 leaves and the corresponding intervals.

Theorems & Definitions (7)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.3