An induced packing of cycles in a graph $G$ is a set of vertex-disjoint cycles such that $G$ has no edge between distinct cycles of the set. The classic Erdős-Pósa theorem asserts that for every positive integer $k$, every graph contains $k$ vertex-disjoint cycles or a set of $O(kcdotlog k)$ vertices intersecting every cycle of $G$. We generalise the classic Erdős-Pósa theorem to the induced packings of cycles of length at least $ell$ for any integer $ell$. We show that there exists a function $f(k,ell)=O(ellcdot kcdotlog k)$ such that for all positive integers $k$ and $ellgeq3$, every graph $G$ contains an induced packing of $k$ cycles of length at least $ell$ or a set $X$ of at most $f(k,ell)$ vertices such that the closed neighbourhood of $X$ intersects every cycle of length at least $ell$ in $G$. We also extend the theorem to cycles containing prescribed vertices. For a graph $G$ and a subset $S$ of $V(G)$, an $S$-cycle in $G$ is a cycle containing a vertex in $S$. We show that for every positive integer $k$, every graph $G$, and every subset $S$ of $V(G)$, $G$ contains an induced packing of $k$ $S$-cycles or a set $X$ of at most $O(k^5)$ vertices such that the closed neighbourhood of $X$ intersects every $S$-cycle in $G$. Our proofs are constructive and yield polynomial-time algorithms finding either the induced packing or the set $X$ in both cases. This is joint work with Pascal Gollin, Tony Huynh, and O-joung Kwon.

Jungho Ahn is an Assistant Professor in the Department of Computer Engineering at Inha University, South Korea. His research spans algorithmic and structural graph theory. He received his Ph.D. in Mathematical Sciences from KAIST in 2023 under the supervision of Sang-il Oum, with a dissertation on algorithmic and structural aspects of graph parameters. Prior to joining Inha University, he held positions as a postdoctoral research associate at Durham University and as a research fellow at the Korea Institute for Advanced Study (KIAS).