2013
DOI: 10.1016/j.dam.2012.12.001
|View full text |Cite
|
Sign up to set email alerts
|

Towards optimal kernel for connected vertex cover in planar graphs

Abstract: We study the parameterized complexity of the connected version of the vertex cover problem, where the solution set has to induce a connected subgraph. Although this problem does not admit a polynomial kernel for general graphs (unless NP ⊆ coNP/poly), for planar graphs Guo and Niedermeier [ICALP'08] showed a kernel with at most 14k vertices, subsequently improved by Wang et al. [MFCS'11] to 4k. The constant 4 here is so small that a natural question arises: could it be already an optimal value for this proble… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
1
1

Citation Types

0
4
0

Year Published

2014
2014
2017
2017

Publication Types

Select...
3
2
1

Relationship

2
4

Authors

Journals

citations
Cited by 8 publications
(4 citation statements)
references
References 23 publications
0
4
0
Order By: Relevance
“…We will show that there is a solution S of (G, k), contradicting our assumption. For rules 13 Hence we can assume that there is no (u, v) path in G − S . Note that this implies that V (Q ) ∩ S is equal to {u, y} for Rule 15, {u, y} or {y, v} for Rule 16, and {y 1 , y 2 } or {y 1 , v} for Rule 17.…”
Section: Rule 17mentioning
confidence: 99%
See 1 more Smart Citation
“…We will show that there is a solution S of (G, k), contradicting our assumption. For rules 13 Hence we can assume that there is no (u, v) path in G − S . Note that this implies that V (Q ) ∩ S is equal to {u, y} for Rule 15, {u, y} or {y, v} for Rule 16, and {y 1 , y 2 } or {y 1 , v} for Rule 17.…”
Section: Rule 17mentioning
confidence: 99%
“…In fact it turns out that for a number of problems on planar graphs, including Planar Dominating Set and Planar Feedback Vertex Set, one can get a kernel of size O(k) by general method of protrusion decomposition [9]. However, in this general algorithm the constants hidden in the O notation are very large, and researchers keep working on problem-specific linear kernels with the constants as small as possible [6,14,17,13,12].…”
Section: Introductionmentioning
confidence: 99%
“…Knowing this, further research is done to reduce the leading constant in the linear function describing the kernel size. For example, the kernel of Alber et al was later improved to 67k by Chen et al [5]; the first linear kernel for Planar Connected Vertex Cover was that of size 14k due to Guo and Niedermeier [10] and it was then reduced to 4k by Wang et al [17] and even to 11 3 k by Kowalik et al [13]. Observe that these constants may be crucial: since we deal with NP-complete problems, in order to find an exact solution in the reduced instance, most likely we need exponential time (or at least superpolynomial, because for planar graphs 2 O( √ k) -time algorithms are often possible), and these constants appear in the exponents.…”
Section: Kernelization and Dischargingmentioning
confidence: 99%
“…In what follows, |V | is denoted by n. This problem is closely related with Connected Vertex Cover (CVC in short), where given a graph G = (V, E) and an integer k we ask whether there is a set S ⊆ V of size at most k such that S is a vertex cover (i.e. every edge of G has an endpoint in S) and S induces a connected subgraph of G. The CVC problem has been intensively studied, in particular there is a series of results on kernels for planar graphs [6,13] culminating in the recent 11 3 k kernel [9]. It is easy to see that C is a connected vertex cover iff V − C is a nonseparating independent set.…”
Section: Introductionmentioning
confidence: 99%