The Grid Minor Theorem states that for every planar graph H, there exists a smallest integer f(H) such that every graph with tree-width at least f(H) contains H as a minor. The only known lower bounds on f(H) beyond the trivial bound f(H)≥∣V(H)∣−1 come from the maximum number of disjoint cycles in H. In this paper, we study f(H) for planar graphs H with no two disjoint cycles. We prove that f(H)=∣V(H)∣−1 for every apex-forest H. This result improves a bound of Leaf and Seymour and contains all known large graphs H meeting the trivial lower bound to our knowledge. We also prove that f(H)≤max{23∣V(H)∣−29,∣V(H)∣−1} for every wheel H.
@article{arxiv.2509.09895,
title = {Tree-width of a graph excluding an apex-forest or a wheel as a minor},
author = {Chun-Hung Liu and Youngho Yoo},
journal= {arXiv preprint arXiv:2509.09895},
year = {2025}
}