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%Write your title here: 
\noindent{\bf\Large Erd\"os-P\'osa Property of planar-$H$-minor models with prescribed vertex sets}

%List the authors as follows (you may underline the speaker, or just
%name the speaker, and mention your co-authors at the end, up to you):  
\noindent \underline{O-joung Kwon} -- {\small MTA SZTAKI, Hungarian Academy of Sciences}

\noindent D\'aniel Marx -- {\small MTA SZTAKI, Hungarian Academy of Sciences}



\vspace{2ex}

\noindent{\bf Abstract:} %Here write your abstract

Robertson and Seymour~\cite{RobertsonS1986} proved that for every planar graph $H$, the class of all $H$-expansions, which are the graphs can be contracted to $H$, has the Erd\"os-P\'osa property, and this does not hold for non-planar graphs.
We generalize this result for disjoint $H$-expansions containing vertices from two sets among several prescribed vertex sets.
This generalization is motivated from Mader's $\mathcal{S}$-path theorem~\cite{Mader78}. 

For a graph $G$ and a set $\mathcal{Z}=\{Z_i\subseteq V(G): 1\le i\le m\}$ with $m\ge 2$, 
an $H$-expansion $F$ in $G$ is \emph{$\mathcal{Z}$-connecting} if $V(F)\cap Z_i\neq \emptyset$ and $V(F)\cap Z_j\neq \emptyset$ for some $i\neq j$.
We show that $\mathcal{Z}$-connecting $H$-expansions have the Erd\"os-P\'osa property:
for a positive integer $k$ and a planar graph $H$, there exists a function $f(k,H)$ such that if $G$ is a graph and $\mathcal{Z}=\{Z_i\subseteq V(G): 1\le i\le m\}$ with $m\ge 2$, then either 
\begin{enumerate}
\item $G$ contains $k$ pairwise vertex-disjoint $\mathcal{Z}$-connecting $H$-expansions, or
\item there is a vertex subset $T$ of size at most $f(k,H)$ in $G$ such that $G\setminus T$ contains no $\mathcal{Z}$-connecting $H$-expansions.
\end{enumerate}
We point out that the function $f$ does not depend on the number $m$ of prescribed vertex sets.
It implies the original theorem by Robertson and Seymour by taking $Z_1=Z_2=V(G)$, and further implies that for a vertex set $S\subseteq V(G)$, $H$-expansions intersecting $S$ also have the Erd\"os-P\'osa property by taking $Z_1=S$ and $Z_2=V(G)$, which is similar to the Erd\"os-P\'osa property of cycles intersecting $S$~\cite{KakimuraKM2011, PontecorviW2012}.
To prove it, we use the Rooted Grid Minor theorem developed by Marx, Seymour, and Wollan~\cite{MarxSW2013}, and mainly develop a procedure to find a vertex in a graph of large tree-width, called an irrelavent vertex, whose deletion preserves the minimum size of a packing set for $\mathcal{Z}$-connecting $H$-expansions. 
We also give an example showing that if the number of required intersecting sets is more than two, then the Erd\"os-P\'osa property does not hold. 
This is joint work with D\'aniel Marx.


%If you need a ref section, do it more or less like that : 
\small
\begin{thebibliography}{1}

\bibitem{KakimuraKM2011}
N.~Kakimura, K.~Kawarabayashi, and D.~Marx.
\newblock Packing cycles through prescribed vertices.
\newblock {\em Journal of Combinatorial Theory, Series B}, 101(5):378 -- 381,
  2011.

\bibitem{Mader78}
W.~Mader.
\newblock \"{U}ber die {M}aximalzahl kreuzungsfreier {$H$}-{W}ege.
\newblock {\em Arch. Math. (Basel)}, 31(4):387--402, 1978/79.

\bibitem{MarxSW2013}
D.~Marx, P.~Seymour, and P.~Wollan.
\newblock Rooted grid minors, 2013.
\newblock arXiv:1307.8138.

\bibitem{PontecorviW2012}
M.~Pontecorvi and P.~Wollan.
\newblock Disjoint cycles intersecting a set of vertices.
\newblock {\em Journal of Combinatorial Theory, Series B}, 102(5):1134 -- 1141,
  2012.

\bibitem{RobertsonS1986}
N.~Robertson and P.~Seymour.
\newblock Graph minors. {V}. {E}xcluding a planar graph.
\newblock {\em Journal of Combinatorial Theory, Series B}, 41(1):92 -- 114,
  1986.

\end{thebibliography}


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