A Linear Kernel for Planar Connected Dominating Set

Lokshtanov, Daniel; Mnich, Matthias; Saurabh, Saket

doi:10.1016/j.tcs.2010.10.045

Cited by 50 publications

(12 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result complements several explicit linear kernels on planar graphs for other domination problems such as Dominating Set [2], Edge Dominating Set [14,20], Efficient Dominating Set [14], Connected Dominating Set [13,17], or Total Dominating Set [12]. It is worth mentioning that our constant is considerably smaller than most of the constants provided by these results.…”

Section: Introductionsupporting

confidence: 70%

A linear kernel for planar red–blue dominating set

Garnero

Thilikos

2017

Discrete Applied Mathematics

View full text Add to dashboard Cite

In the Red-Blue Dominating Set problem, we are given a bipartite graph G = (V B ∪ V R , E) and an integer k, and asked whether G has a subset D ⊆ V B of at most k "blue" vertices such that each "red" vertex from V R is adjacent to a vertex in D. We provide the first explicit linear kernel for this problem on planar graphs, of size at most 43k. Rule 4Let v, w be two distinct blue vertices such that |P (v, w)| ≥ 1. Let D = {d ∈ V B : P (v, w) ⊆ N (d)}. We distinguish the following cases:

show abstract

Section: Introductionsupporting

confidence: 70%

A linear kernel for planar red–blue dominating set

Garnero

Thilikos

2017

Discrete Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…On the lower bounds side, Cygan et al [8] have shown that the existence of a size O(k (d−1)(d−3)− ) kernel, > 0, for Dominating Set on d-degenerate graphs would imply NP ⊆ coNP/poly. For the Connected Dominating Set problem linear kernels are only known for planar [29] and H-topological-minor-free graphs [19]. Polynomial kernels are excluded already for graphs of bounded degeneracy [9], assuming NP ⊆ coNP/poly.…”

Section: Introductionmentioning

confidence: 99%

Lossy Kernels for Connected Dominating Set on Sparse Graphs

Eiben¹,

Kumar²,

Mouawad³

et al. 2019

SIAM J. Discrete Math.

View full text Add to dashboard Cite

For α > 1, an α-approximate (bi-)kernel is a polynomial-time algorithm that takes as input an instance (I, k) of a problem Q and outputs an instance (I , k ) (of a problem Q ) of size bounded by a function of k such that, for every c ≥ 1, a c-approximate solution for the new instance can be turned into a (c · α)-approximate solution of the original instance in polynomial time. This framework of lossy kernelization was recently introduced by Lokshtanov et al. We study Connected Dominating Set (and its distance-r variant) parameterized by solution size on sparse graph classes like biclique-free graphs, classes of bounded expansion, and nowhere dense classes. We prove that for every α > 1, Connected Dominating Set admits a polynomial-size α-approximate (bi-)kernel on all the aforementioned classes. Our results are in sharp contrast to the kernelization complexity of Connected Dominating Set, which is known to not admit a polynomial kernel even on 2-degenerate graphs and graphs of bounded expansion, unless NP ⊆ coNP/poly. We complement our results by the following conditional lower bound. We show that if a class C is somewhere dense and closed under taking subgraphs, then for some value of r ∈ N there cannot exist an α-approximate bi-kernel for the (Connected) Distance-r Dominating Set problem on C for any α > 1 (assuming the Gap Exponential Time Hypothesis).

show abstract

“…We find it a bit surprising that the connected version of FVS was not considered in the literature until now. This is in complete contrast to the fact that the connected variant of other problems like Vertex Cover-Connected Vertex Cover [13,18,23,27], Dominating Set-Connected Dominating Set [16,20,21,25] are extremely well studied in the literature. In this paper, we continue the algorithmic study of CFVS from the view-point of parameterized algorithms.…”

Section: Introductionmentioning

confidence: 90%

FPT Algorithms for Connected Feedback Vertex Set

Misra

Philip

Raman

et al. 2010

WALCOM: Algorithms and Computation

Self Cite

View full text Add to dashboard Cite

Abstract. We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G = (V, E) and an integer k, decide whether there exists) on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses as subroutine, a parameterized algorithm for Group Steiner Tree, a well studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree of independent interest and believe that it will be useful for obtaining parameterized algorithms for other connectivity problems.

show abstract

A Linear Kernel for Planar Connected Dominating Set

Cited by 50 publications

References 18 publications

A linear kernel for planar red–blue dominating set

A linear kernel for planar red–blue dominating set

Lossy Kernels for Connected Dominating Set on Sparse Graphs

FPT Algorithms for Connected Feedback Vertex Set

Contact Info

Product

Resources

About