An Induced $A$-Path Theorem

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.