# Parameterized Complexity Dichotomy for (r, ℓ)-Vertex Deletion

Parameterized Complexity Dichotomy for (r, ℓ)-Vertex Deletion For two integers r, ℓ ≥ 0, a graph G = (V, E) is an (r, ℓ)-graph if V can be partitioned into r independent sets and ℓ cliques. In the parameterized (r, ℓ)-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, ℓ)-graph. This problem is NP-hard if r + ℓ ≥ 1 and encompasses several relevant problems such as Vertex Cover and Odd Cycle Transversal. The parameterized complexity of (r, ℓ)-Vertex Deletion was known for all values of (r, ℓ) 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, ℓ)-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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Theory of Computing Systems Springer Journals

# Parameterized Complexity Dichotomy for (r, ℓ)-Vertex Deletion

, Volume 61 (3) – Oct 27, 2016
18 pages

/lp/springer_journal/parameterized-complexity-dichotomy-for-r-vertex-deletion-sQRhGDq9AY
Publisher
Springer US
Subject
Computer Science; Theory of Computation
ISSN
1432-4350
eISSN
1433-0490
D.O.I.
10.1007/s00224-016-9716-y
Publisher site
See Article on Publisher Site

### Abstract

For two integers r, ℓ ≥ 0, a graph G = (V, E) is an (r, ℓ)-graph if V can be partitioned into r independent sets and ℓ cliques. In the parameterized (r, ℓ)-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, ℓ)-graph. This problem is NP-hard if r + ℓ ≥ 1 and encompasses several relevant problems such as Vertex Cover and Odd Cycle Transversal. The parameterized complexity of (r, ℓ)-Vertex Deletion was known for all values of (r, ℓ) 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, ℓ)-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.

### Journal

Theory of Computing SystemsSpringer Journals

Published: Oct 27, 2016

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just \$49/month

### Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

### Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

### Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

### Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

DeepDyve

DeepDyve

### Pro

Price

FREE

\$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations