Tree-width of hypergraphs and surface duality

TL;DR

This paper extends the understanding of tree-width relationships between hypergraphs and their duals on surfaces, providing bounds that depend on surface genus and hyperedge size, generalizing planar graph duality.

Contribution

It proves a new bound on the tree-width of hypergraph duals on surfaces of genus k, generalizing the planar case to higher genus surfaces.

Findings

Tree-width of hypergraph duals is bounded by surface genus and maximum hyperedge size.

Generalizes planar graph duality results to higher genus surfaces.

Provides a new framework for analyzing hypergraph duality on complex surfaces.

Abstract

In Graph Minor III, Robertson and Seymour conjecture that the tree-width of a planar graph and that of its dual differ by at most one. We prove that given a hypergraph H on a surface of Euler genus k, the tree-width of H^* is at most the maximum of tw(H) + 1 + k and the maximum size of a hyperedge of H^*.