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

TL;DR

The paper links minor exclusions of apex-typed graph families to tight linear bounds on layered and focused decompositions: excluding an apex-forest minor yields a bound $ ext{lpw}(G) extle |V(H)|-2$ and excluding an apex-linear forest minor yields $ ext{ltd}(G) extle |V(H)|-2$, with optimality. It introduces focused variants and a central lemma that connects the absence of $S$-rooted $P_ heta$ models to linear bounds on $ ext{td}(G,S)$, enabling a linear bound for the $S$-focused treedepth. The authors refine these results to stronger, diameter- or radius-aware forms and extend the approach to fans and related fan-like apices via separation-based techniques, culminating in tight lower-bound constructions that confirm optimality. The results advance the understanding of how forbidden apex minors constrain layered decompositions and have implications for related graph-structure parameters and minor-closed classes.

Abstract

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~$S$.

Figures (1)

• Figure 1: An outline of the proof of \ref{['lem:main']}. Note that the notation is consistent with the actual proof that we give. The set $X$ represents vertices that were selected as attachments in $J$ at higher levels of induction and must therefore not be used as attachments in lower levels of induction (\ref{['item:X-degree']}). Suppose that we work with $\ell = 7$. The vertex $x_0$ is the lowest common ancestor of $S$ in $T$. In the general case, we split the descendants of $x_0$ into two groups: $A_1$ and $A_2$ both containing elements of $S$. Suppose that (as in the figure), there are $a_1 = 4$ pairwise vertex-disjoint paths connecting $S \cap A_1$ and $V(P_T(x_0)) \setminus X$. Then, we can add $X_2' = \{x_{1,1},x_{1,2},x_{1,3},x_{1,4}\}$ to $X$ and call induction for $X_2'$, $S_2' = S \cap A_2$, and $\ell-a_'$. To this end, we need to assume that $\mathop{\mathrm{td}}\nolimits(G-X_2',S_2') \geqslant 2(\ell-a_1)$. If this holds, then by induction, we obtain a $(G,S_2',T)$-path of order $\ell-a_1 = 3$ that avoids $X_2'$. On the right side of the figure, we show how we combine all these to obtain a $(G,S,T)$-path of order $\ell = 7$. When $\mathop{\mathrm{td}}\nolimits(G-X_2',S_2') < 2(\ell-a_1)$ we can try to argue symmetrically. If this fails, it means that $\mathop{\mathrm{td}}\nolimits(G-X_1',S_1') < 2(\ell-a_2)$. Having both, we argue that $\mathop{\mathrm{td}}\nolimits(G-X,S) < 2\ell$, which will be a contradiction. The argument is intuitively simple. We assume $a_1 \leqslant a_2$ and we eliminate vertices of $X_1 \cup Y_1$ where $Y_1$ is a set of $a_1$ elements separating $A_1 \cap S$ from $V(R) \setminus X$. It suffices to argue that for every connected component $C$ of $G - (X \cup X_1 \cup Y_1)$, we have $\mathop{\mathrm{td}}\nolimits(C, S \cap V(C)) < 2\ell - 2a_1$.

Theorems & Definitions (37)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Corollary 5
• Theorem 6
• Theorem 7
• Lemma 9
• proof : Proof of \ref{['lem:main']}.
• Claim 10