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%Write your title here: 
\noindent{\bf\Large FPT results through potential
maximal cliques}

%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 Fedor V. Fomin -- {\small University of Bergen}

\noindent Mathieu Liedloff -- {\small Universit\'e d'Orl\'eans}

\noindent \underline{Pedro Montealegre} -- {\small Universit\'e d'Orl\'eans}

\noindent Ioan Todinca-- {\small Universit\'e d'Orl\'eans}


\vspace{2ex}

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

In many graph problems, like  \textsc{Longest Induced Path}, \textsc{Maximum Induced Forest}, etc., we are given as input a graph $G$ and the goal is to compute a largest induced subgraph $G[F]$, of treewidth at most a constant $t$, and satisfying some property $\mathcal{P}$. 
Fomin et al.~\cite{FoToVi14} proved that this generic problem is polynomial on the class of graphs ${\mathcal{G}}_{\poly}$, i.e., the graphs having at most $\poly(n)$ minimal separators for some polynomial $\poly$, when property $\cP$ is expressible in counting monadic second order logic (CMSO).  The algorithm is based on the enumeration of potential maximal cliques.

Here we extend this result in two directions:
\begin{itemize}
\item The generic problem can be solved in time $\cO^*(4^{\vc})$ or $\cO^*(\pmcb^{\mw})$, where $\vc$ and $\mw$ correspond to the \emph{vertex cover} and the \emph{modular width} of the input graph.



\item Consider the class $\Gpolyk$, formed by graphs of $\Gpoly$ to which we may add a set of at most $k$ vertices with arbitrary adjacencies, called \emph{modulator}. We prove that the generic optimization problem is fixed parameter tractable on $\Gpolyk$, with parameter $k$, if the modulator is also part of the input. 

\end{itemize} 

%This paper is about ... Cite like that \cite{CCHPP}, \cite{Wol15}. 


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\bibitem{FoToVi14}
Fedor~V. Fomin, Ioan Todinca, and Yngve Villanger.
\newblock Large induced subgraphs via triangulations and {CMSO}.
\newblock In SIAM, editor, {\em Proceedings of SODA 2014}, 2014.
\newblock See also http://arxiv.org/abs/1309.1559.

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