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Parameterized Algorithms for Deletion to (r,l)-graphs

For fixed integers $r,\ell \geq 0$, a graph $G$ is called an {\em $(r,\ell)$-graph} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\ell$ cliques. This brings us to the following natural parameterized questions: {\sc Vertex $(r,\ell)$-Partization} and {\sc Edge $(r,\ell)$-Partization}. An input to these problems consist of a graph $G$ and a positive integer $k$ and the objective is to decide whether there exists a set $S\subseteq V(G)$ ($S\subseteq E(G)$) such that the deletion of $S$ from $G$ results in an $(r,\ell)$-graph. These problems generalize well studied problems such as {\sc Odd Cycle Transversal}, {\sc Edge Odd Cycle Transversal}, {\sc Split Vertex Deletion} and {\sc Split Edge Deletion}. We do not hope to get parameterized algorithms for either {\sc Vertex $(r,\ell)$-Partization} or {\sc Edge $(r,\ell)$-Partization} when either of $r$ or $\ell$ is at least $3$ as the recognition problem itself is NP-complete. This leaves the case of $r,\ell \in \{1,2\}$. We almost complete the parameterized complexity dichotomy for these problems. Only the parameterized complexity of {\sc Edge $(2,2)$-Partization} remains open. We also give an approximation algorithm and a Turing kernelization for {\sc Vertex $(r,\ell)$-Partization}. We use an interesting finite forbidden induced graph characterization, for a class of graphs known as $(r,\ell)$-split graphs, properly containing the class of $(r,\ell)$-graphs. This approach to obtain approximation algorithms could be of an independent interest.