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%Write your title here: 
\noindent{\bf\Large Parameterized Complexity Dichotomy for $(r,\ell)$-\textsc{Vertex Deletion}}

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\noindent \underline{J. Baste} -- {\small LIRMM, Université de Montpellier, Montpellier, France}

\noindent L. Faria -- {\small FFP, Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil}
                         

\noindent S. Klein -- {\small Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil}

\noindent I. Sau -- {\small LIRMM, CNRS, Montpellier, France}



\vspace{2ex}

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


For two integers $r, \ell \geq 0$, a graph $G = (V, E)$ is an \emph{$(r,\ell)$-graph} if $V$ can be partitioned into $r$ independent sets and $\ell$ cliques. In the parameterized $(r,\ell)$-\textsc{Vertex Deletion} problem, given a graph $G$ and an integer $k$, one has to decide whether at most $k$ vertices can be removed from $G$ to obtain an $(r,\ell)$-graph. This problem is {\sf NP}-hard if $r+\ell \geq 1$ and encompasses several relevant problems such as \textsc{Vertex Cover} and \textsc{Odd Cycle Transversal}. The parameterized complexity of $(r,\ell)$-\textsc{Vertex Deletion} was known for all values of $(r,\ell)$ except for $(2,1)$, $(1,2)$, and $(2,2)$. We prove that each of these three cases is {\sf FPT} and, furthermore, solvable in single-exponential time, which is asymptotically optimal in terms of $k$. We consider as well the version of $(r,\ell)$-\textsc{Vertex Deletion} where the set of vertices to be removed has to induce an independent set, and provide also a parameterized complexity dichotomy for this problem.



% Let $r$ and $l$ be two fixed integers. We say that $G = (V,E)$ is an \emph{($r$,$\ell$)-graph} if $V$ can be partitioned into $r$ independent sets and $\ell$ cliques. Given a graph $G=(V,E)$ and an integer $k$, the \textsc{($r$, $\ell$)-Vertex Deletion} problem parameterized by $k$ 
% consist of deciding if we can remove at most $k$ vertices from $G$ such that the resulting graph is an $(r,\ell)$-graph.
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% This paper is about ... Cite like that \cite{CCHPP}, \cite{Wol15}. 


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% \bibitem{CCHPP}
% R.~H. Chitnis, M.~Cygan, M.~Hajiaghayi, M.~Pilipczuk, and M.~Pilipczuk.
% \newblock Designing {FPT} algorithms for cut problems using randomized
%   contractions.
% \newblock In {\em 53rd Annual {IEEE} Symposium on Foundations of Computer
%   Science (FOCS)}, pages 460--469, 2012.

% \bibitem{Wol15}
% P.~Wollan.
% \newblock The structure of graphs not admitting a fixed immersion.
% \newblock {\em Journal of Combinatorial Theory, Series {B}}, 110:47--66, 2015.

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