Tree-width of hypergraphs and surface duality

TL;DR

This paper generalizes a known relationship between the tree-width of planar graphs and their duals to hypergraphs embedded on arbitrary surfaces, providing a tight bound and a formal proof.

Contribution

It extends the duality relationship of tree-width from planar graphs to hypergraphs on general surfaces, with a proven tight bound.

Findings

Tree-width of hypergraphs and their duals differ by at most one on general surfaces

The bound on the difference is proven to be tight

Generalizes previous planar graph results to hypergraphs on surfaces

Abstract

In Graph Minors III, Robertson and Seymour write: "It seems that the tree-width of a planar graph and the tree-width of its geometric dual are approximately equal - indeed, we have convinced ourselves that they differ by at most one". They never gave a proof of this. In this paper, we prove a generalisation of this statement to embedding of hypergraphs on general surfaces, and we prove that our bound is tight.