Excluding an apex-forest or a fan as quickly as possible

We show that every graph $G$ excluding an apex-forest $H$ as a minor has layered pathwidth at most $|V(H)|-2$, and that every graph $G$ excluding an apex-linear forest (such as a fan) $H$ as a minor has layered treedepth at most $|V(H)|-2$. We further show that both bounds are optimal. These results improve on recent results of Hodor, La, Micek, and Rambaud (2025): The first result improves the previous best-known bound by a multiplicative factor of $2$, while the second strengthens a previous quadratic bound. In addition, we reduce from quadratic to linear the bound on the $S$-focused treedepth $\mathrm{td}(G,S)$ for graphs $G$ with a prescribed set of vertices $S$ excluding models of paths in which every branch set intersects~$S$.

💡 Research Summary

This paper investigates the relationship between excluded minors of apex‑type graphs and two layered graph parameters: layered pathwidth and layered treedepth. The authors improve upon recent bounds by Hodor, La, Micek, and Rambaud (2025), providing optimal linear bounds that are tight up to an additive constant.

The first main result concerns apex‑forests. An apex‑forest H is a graph that becomes a forest after the removal of a single vertex. The authors prove that for any apex‑forest H with at least three vertices, every H‑minor‑free graph G satisfies
  lpw(G) ≤ |V(H)| – 2,
where lpw denotes layered pathwidth. Previously the best known bound was f(k)=2k–3 for k=|V(H)|, so the new result halves the multiplicative constant. The proof constructs a layering L and a path decomposition W simultaneously, ensuring that each bag intersects each layer in at most |V(H)|–2 vertices. The argument exploits the structural property that removing an apex vertex from any H‑minor‑free graph yields a forest, which limits the number of vertices that can appear in any single layer.

The second main result deals with apex‑linear forests, a class that includes fans (a path plus a universal vertex). An apex‑linear forest H becomes a subgraph of a path after deleting one vertex. The authors show that for any such H with at least three vertices, every H‑minor‑free graph G satisfies
  ltd(G) ≤ |V(H)| – 2,
where ltd denotes layered treedepth. Earlier work gave a quadratic bound of ⌈|V(H)|/2⌉. The new proof builds an elimination forest F together with a layering L, guaranteeing that each subtree of F intersects each layer in at most |V(H)|–2 vertices. The key insight is that the apex vertex forces a hierarchical decomposition where each level can contain only a limited number of vertices without creating an H‑minor.

A third contribution concerns “focused” parameters, where a distinguished vertex set S ⊆ V(G) is treated as important. The authors answer an open question from the previous work by proving a linear bound for S‑focused treedepth. Specifically, if a graph G contains no S‑rooted model of the ℓ‑vertex path Pℓ, then
  td(G, S) ≤ 2ℓ – 2,
where td(G, S) denotes the S‑focused treedepth. The proof introduces a novel (G, S, T)‑path structure based on a depth‑first search tree T, consisting of a “spine” and ℓ vertex‑disjoint “ribs” that connect the spine to S. A technical lemma shows that if the focused treedepth of G−X (relative to S) is at least 2ℓ−1, then a suitable (G, S, T)‑path of order ℓ can be built while avoiding a prescribed set X of vertices as attachments. Applying this lemma with X=∅ yields the desired linear bound.

The authors also demonstrate optimality of the bounds. In Section 7 they construct families of H‑minor‑free graphs whose layered pathwidth or layered treedepth exactly equals |V(H)|–2, showing that the constants cannot be improved. Moreover, they derive corollaries linking the parameters to the radius of a connected graph: for any apex‑linear forest H and any H‑minor‑free connected graph G,
  td(G) ≤ (|V(H)|–2)·rad(G) + 1,
and similarly for pathwidth with the same multiplicative factor.

Overall, the paper provides a clean, tight characterization of how excluding an apex‑forest or a fan (or any apex‑linear forest) controls layered decompositions. By reducing previously quadratic or larger bounds to optimal linear ones, it opens the door to stronger applications in clustered coloring, algorithmic graph theory, and the study of minor‑closed classes where layered parameters play a crucial role.