Separating Notions of Graph Width: the Adaptive, Normal, Linear, Entropic, and Submodular Width

2026-09-30

AI Summary

This study addresses the long-standing lack of unified separation among five width concepts in graph theory: adaptive, normal, linear, entropic, and submodular. To resolve this, the authors construct an explicit 32-vertex counterexample graph. They derive upper bounds using the Ingleton and Zhang–Yeung inequalities, while establishing lower bounds through the construction of modular, normal, and non-entropic polyhedra, thereby precisely computing all width values except the entropic width. The primary contribution is the first pairwise separation of these five widths within a single graph instance, filling a critical theoretical gap in their distinguishability. Furthermore, by obtaining exact values or tight bounds for each width measure, this work effectively validates related theoretical hypotheses concerning the structural distinctions among these fundamental graph-theoretic parameters.

Abstract

We describe one explicit simple graph G on 32 vertices whose adaptive, normal, linear, entropic, and submodular widths are pairwise distinct. We compute all these widths exactly, except for the entropic width, where we only give a lower and upper bound. We use Ingleton's inequality and the Zhang-Yeung inequality for upper bounds, and give explicit constructions of modular, normal, linear, entropic, and non-entropic polymatroids for lower bounds.

Problem

• graph width
• entropic width
• submodular width
• polymatroid
• Ingleton's inequality

Innovation

• Graph Width
• Polymatroids
• Entropic Width
• Ingleton's Inequality
• Zhang-Yeung Inequality