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A minor-model of a graph $H$ in a graph $G$ is a subgraph of $G$ that can be contracted to $H$. We prove that for a positive integer $ell$ and a non-empty planar graph $H$ with at least $ell-1$ connected components, there exists a function $f_{H, ell}:mathbb{N}rightarrow mathbb{R}$ satisfying the property that every graph $G$ with a family of vertex subsets $Z_1, ldots, Z_m$ contains either $k$ pairwise vertex-disjoint minor-models of $H$ each intersecting at least $ell$ sets among prescribed vertex sets, or a vertex subset of size at most $f_{H, ell}(k)$ that meets all such minor-models of $H$. This function $f_{H, ell}$ is independent with the number $m$ of given sets, and thus, our result generalizes Maders $mathcal S$-path Theorem, by applying $ell=2$ and $H$ to be the one-vertex graph. We prove that such a function $f_{H, ell}$ does not exist if $H$ consists of at most $ell-2$ connected components.
A Group Labeled Graph is a pair $(G,Lambda)$ where $G$ is an oriented graph and $Lambda$ is a mapping from the arcs of $G$ to elements of a group. A (not necessarily directed) cycle $C$ is called non-null if for any cyclic ordering of the arcs in $C$
Robertson and Seymour proved that the family of all graphs containing a fixed graph $H$ as a minor has the ErdH{o}s-Posa property if and only if $H$ is planar. We show that this is no longer true for the edge version of the ErdH{o}s-Posa property, an
A chordless cycle, or equivalently a hole, in a graph $G$ is an induced subgraph of $G$ which is a cycle of length at least $4$. We prove that the ErdH{o}s-Posa property holds for chordless cycles, which resolves the major open question concerning th
We prove that there exists a function $f:mathbb{N}rightarrow mathbb{R}$ such that every digraph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a vertex set $X$ of size at most $f(k)$ hitt
We prove that there exists a function $f(k)=mathcal{O}(k^2 log k)$ such that for every $C_4$-free graph $G$ and every $k in mathbb{N}$, $G$ either contains $k$ vertex-disjoint holes of length at least $6$, or a set $X$ of at most $f(k)$ vertices such