Quickly excluding an apex-forest

TL;DR

The paper studies how forbidding apex-type minors interacts with layered width parameters, proving that every $X$-minor-free graph has bounded layered pathwidth when $X$ is an apex-forest with $|V(X)|\ge 2$, with an explicit bound $\text{lpw}(G) \le 2|V(X)|-3$. The authors introduce rooted width parameters $pw(G,S)$ and $td(G,S)$ and develop a suite of decompositions and separation lemmas that extend forest-exclusion bounds to rooted models, plus a focused grid-minor/tangle framework for treewidth. They further derive diameter-dependent bounds and establish dualities between treewidth and tangles in the rooted setting, leading to Erdős–Pósa results for rooted models of plane graphs and trees. Together, these results not only improve prior bounds but also provide new structural tools for deeper minor-closed graph theory questions and related algorithmic applications.

Abstract

We give a short proof that for every apex-forest $X$ on at least two vertices, graphs excluding $X$ as a minor have layered pathwidth at most $2|V(X)|-3$. This improves upon a result by Dujmović, Eppstein, Joret, Morin, and Wood (SIDMA, 2020). Our main tool is a structural result about graphs excluding a forest as a rooted minor, which is of independent interest. We develop similar tools for treedepth and treewidth. We discuss implications for Erdős-Pósa properties of rooted models of minors in graphs.

Figures (11)

• Figure 1: The green and blue bags depict a path decomposition of $H$, an induced subgraph of $G$ such that $S \subseteq V(H)$. Each component of $G - V(H)$ (yellow) has all the neighbors in one of the bags. Such a bag does not have to be unique.
• Figure 2: Illustration of the proof of \ref{['lemma:key_lemma_wgood']}. On the left, we depict the initial situation and on the right, we depict the result of applying the procedure from the lemma. Bags of path decompositions (of $(A',S \cap V(A'))$ on the left and of $(P,S \cap V(P))$ on the right) are green and blue alternately. In yellow, we show the components that are left after removing all vertices of respective path decompositions. The latter is obtained from the former by contracting $|V(P) \cap V(Q)|$ disjoint $V(P)$--$V(B')$ paths (in blue).
• Figure 3: An illustration of the proof of \ref{['lemma:find_wTS_spanning_separation']}. We consider $F$ to be a forest ($|V(F)| = 5$ in the figure). In pink, we depict the branch sets of the rooted model of $F - t$. We argue that if $|V(P) \cap V(Q)| < |V(F)|$, then $(P,Q)$ contradicts the maximality of $(A^0,B^0)$. Hence, $V(A)$ is connected with $V(B')$ by $5$ pairwise disjoint paths. We add the blue branch set containing $v$ to the model and extend pink branch sets using the paths obtaining a $(V(A') \cap V(B'))$-rooted model of $F$ in $A'$.
• Figure 4: An illustration of the proof of \ref{['thm:Spw']}. On the left, we depict the situation, where $|V(P)\cap V(Q)| < |V(F)|$. We can extend the path decomposition by appending the bag $(V(A) \cap V(B)) \cup (V(P) \cap V(Q))$ (the last green bag in the figure). On the right, we depict the opposite situation, where $|V(P)\cap V(Q)| \geqslant |V(F)|$. Then, we simply extend the model and make it $S$-rooted.
• Figure 5: An illustration of how we construct the layering $(L_j \mid j \geqslant 0)$ in the proof of \ref{['lemma:lpw_technical']}.

Theorems & Definitions (51)

• Theorem 1
• Theorem 2
• Corollary 3
• Corollary 4
• Theorem 5: Menger's Theorem
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10: based on Marx2017