Polyhedral graph abstractions and an approach to the Linear Hirsch Conjecture

TL;DR

This paper introduces a new combinatorial framework for polyhedral graphs, providing tools to analyze their diameters and offering a potential approach to disprove the Linear Hirsch Conjecture.

Contribution

It presents a novel abstraction that captures polyhedral graph properties and explores variants with different diameter bounds, including a method to challenge the Linear Hirsch Conjecture.

Findings

One variant matches the best known diameter bounds.

Another variant exhibits superlinear diameter growth.

Provides a concrete approach to disprove the Linear Hirsch Conjecture.

Abstract

We introduce a new combinatorial abstraction for the graphs of polyhedra. The new abstraction is a flexible framework defined by combinatorial properties, with each collection of properties taken providing a variant for studying the diameters of polyhedral graphs. One particular variant has a diameter which satisfies the best known upper bound on the diameters of polyhedra. Another variant has superlinear asymptotic diameter, and together with some combinatorial operations, gives a concrete approach for disproving the Linear Hirsch Conjecture.