# Parameterized complexity dichotomy for $(r,\ell)$-Vertex Deletion

1 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : For two integers $r, \ell \geq 0$, a graph $G = (V, E)$ is an $(r,\ell)$-graph if $V$ can be partitioned into $r$ independent sets and $\ell$ cliques. In the parameterized $(r,\ell)$-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 NP-hard if $r+\ell \geq 1$ and encompasses several relevant problems such as Vertex Cover and Odd Cycle Transversal. The parameterized complexity of $(r,\ell)$-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 FPT and, furthermore, solvable in single-exponential time, which is asymptotically optimal in terms of $k$. We consider as well the version of $(r,\ell)$-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.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal-lirmm.ccsd.cnrs.fr/lirmm-01272704
Contributor : Ignasi Sau <>
Submitted on : Thursday, February 11, 2016 - 11:56:28 AM
Last modification on : Thursday, March 5, 2020 - 6:55:56 PM

### Identifiers

• HAL Id : lirmm-01272704, version 1
• ARXIV : 1504.05515

### Citation

Julien Baste, Luerbio Faria, Sulamita Klein, Ignasi Sau Valls. Parameterized complexity dichotomy for $(r,\ell)$-Vertex Deletion. 2016. ⟨lirmm-01272704⟩

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