Erd\H{o}s-P\'osa property of $A$-paths in unoriented group-labelled graphs

Abstract

We characterize the obstructions to the Erd\H{o}s-P\'osa property of $A$-paths in unoriented group-labelled graphs. As a result, we prove that for every finite abelian group $\Gamma$ and for every subset $\Lambda$ of $\Gamma$, the family of $\Gamma$-labelled $A$-paths whose lengths are in $\Lambda$ satisfies the half-integral Erd\H{o}s-P\'osa property. Moreover, we give a characterization of such $\Gamma$ and $\Lambda\subseteq\Gamma$ for which the same family of $A$-paths satisfies the full Erd\H{o}s-P\'osa property.