An Induced $A$-Path Theorem
Robert Hickingbotham, Gwenaël Joret
TL;DR
The paper extends Gallai's $\mathcal{A}$-path theorem to induced settings by proving that for any graph $G$ and $\mathcal{A}\subseteq V(G)$, either there exist $k$ pairwise anti-complete $\mathcal{A}$-paths or a vertex set $Z$ with $|Z|\le 78(k-1)$ such that removing the closed neighbourhood $N_G[Z]$ destroys all $\mathcal{A}$-paths. It strengthens this with an induced variant using balls of radius $4$, giving $|Z|\le 4(k-1)$, and extends the results to long induced $\mathcal{A}$-paths. The proof leverages the frame technique with $\mathcal{A}$-frames and $\ell$-hub-trees, employing induction on $k$ to obtain the desired dichotomy and the small separators. These results contribute to coarse graph theory and have potential applications to induced minors and the Induced Menger Conjecture, and are framed as steps toward an Induced Minor Structure Theorem.
Abstract
Given a graph $G$ and $\mathcal{A}\subseteq V(G)$, a classical theorem of Gallai (1964) states that for every positive integer $k$, the graph $G$ contains $k$ pairwise vertex-disjoint $\mathcal{A}$-paths, or a set $Z\subseteq V(G)$ of size at most $2(k-1)$ such that $G-Z$ contains no $\mathcal{A}$-paths. We generalise Gallai's theorem to the induced setting: We prove that $G$ contains $k$ pairwise anti-complete $\mathcal{A}$-paths, or a set $Z$ of size at most $78(k-1)$ such that, after removing the closed neighbourhood of $Z$, the resulting graph has no $\mathcal{A}$-path. Here, two paths are anti-complete if they are vertex disjoint and there is no edge in $G$ having one endpoint in each of them. We further show that the bound $78(k-1)$ on the size of $Z$ can be reduced to $4(k-1)$ if one removes the balls of radius $4$ around the vertices of $Z$ (instead of radius $1$), which is within a factor $2$ of optimal. We also establish analogous results for long induced $\mathcal{A}$-paths.