Polynomial dependencies in the Graph Minors Structural Theorem (GMST)

Establish polynomial bounds on the parameter dependencies in the Graph Minors Structural Theorem of Robertson–Seymour, thereby avoiding exponential blow-ups and improving the gap functions in this paper’s parametric equivalences from exponential in k and h toward polynomial bounds.

Background

The paper’s exponential gap functions partly arise from exponential dependencies in the GMST bounds used in their technical development (e.g., in Lemma 4.2 and related vortex-splitting operations).

The authors suggest that polynomially bounding these dependencies would be a major advancement and would likely remove a primary source of exponentiation in their proofs.