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- W813383208 abstract "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 $Ssubseteq V(G)$ ($Ssubseteq E(G)$) 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." @default.
- W813383208 created "2016-06-24" @default.
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- W813383208 date "2015-04-30" @default.
- W813383208 modified "2023-09-27" @default.
- W813383208 title "Parameterized Algorithms for Deletion to (r,l)-graphs" @default.
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