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.

💡 Research Summary

The paper presents a substantial extension of Gallai’s classic A‑path theorem to the induced‑graph setting, introducing the notion of anti‑complete A‑paths—paths that are vertex‑disjoint and have no edges joining the two paths. The authors prove two main results. Theorem 3 states that for any integer k, a graph G either contains k pairwise anti‑complete A‑paths or there exists a vertex set Z of size at most 78(k − 1) such that after deleting the closed neighbourhood N_G