Towards optimal kernel for connected vertex cover in planar graphs

Kowalik, Łukasz; Pilipczuk, Marcin; Suchan, Karol

doi:10.1016/j.dam.2012.12.001

Cited by 8 publications

(4 citation statements)

References 23 publications

(33 reference statements)

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“…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%

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A $$14k$$ -Kernel for Planar Feedback Vertex Set via Region Decomposition

Bonamy

Kowalik

2014

Parameterized and Exact Computation

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We show a kernel of at most 13k vertices for the Feedback Vertex Set problem restricted to planar graphs, i.e., a polynomial-time algorithm that transforms an input instance (G, k) to an equivalent instance with at most 13k vertices. To this end we introduce a few new reduction rules. However, our main contribution is an application of the region decomposition technique in the analysis of the kernel size. We show that our analysis is tight, up to a constant additive term. * A preliminary version (with a slightly weaker result) was presented at IPEC 2014,

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Section: Rule 17mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A $$14k$$ -Kernel for Planar Feedback Vertex Set via Region Decomposition

Bonamy

Kowalik

2014

Parameterized and Exact Computation

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“…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%

Nonblocker in H-minor free graphs: kernelization meets discharging

Kowalik¹

2012

Preprint

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Perhaps the best known kernelization result is the kernel of size 335k for the Planar Dominating Set problem by Alber et al. [1], later improved to 67k by Chen et al. [5]. This result means roughly, that the problem of finding the smallest dominating set in a planar graph is easy when the optimal solution is small. On the other hand, it is known that Planar Dominating Set parameterized by k ′ = |V | − k (also known as Planar Nonblocker) has a kernel of size 2k ′ . This means that Planar Dominating Set is easy when the optimal solution is very large. We improve the kernel for Planar Nonblocker to 7 4 k ′ . This also implies that Planar Dominating Set has no kernel of size at most ( 73 −ǫ)k, for any ǫ > 0, unless P = NP. This improves the previous lower bound of (2 − ǫ)k of [5]. Both of these results immediately generalize to H-minor free graphs (without changing the constants).In our proof of the bound on the kernel size we use a variant of the discharging method (used e.g. in the proof of the four color theorem). We give some arguments that this method is natural in the context of kernelization and we hope it will be applied to get improved kernel size bounds for other problems as well.As a by-product we show a result which might be of independent interest: every n-vertex graph with no isolated vertices and such that every pair of degree 1 vertices is at distance at least 5 and every pair of degree 2 vertices is at distance at least 2 has a dominating set of size at most 3 7 n.

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“…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%

A 9k Kernel for Nonseparating Independent Set in Planar Graphs

Kowalik

Mucha

2012

Graph-Theoretic Concepts in Computer Science

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We study kernelization (a kind of efficient preprocessing) for NP-hard problems on planar graphs. Our main result is a kernel of size at most 9k vertices for the Planar Maximum Nonseparating Independent Set problem. A direct consequence of this result is that Planar Connected Vertex Cover has no kernel with at most (9/8 − ǫ)k vertices, for any ǫ > 0, assuming P = NP. We also show a very simple 5k-vertices kernel for Planar Max Leaf, which results in a lower bound of (5/4 − ǫ)k vertices for the kernel of Planar Connected Dominating Set (also under P = NP).As a by-product we show a few extremal graph theory results which might be of independent interest. We prove that graphs that contain no separator consisting of only degree two vertices contain (a) a spanning tree with at least n/4 leaves and (b) a nonseparating independent set of size at least n/9 (also, equivalently, a connected vertex cover of size at most 8 9 n). The result (a) is a generalization of a theorem of Kleitman and West [8] who showed the same bound for graphs of minimum degree three. Finally we show that every n-vertex outerplanar graph contains an independent set I and a collection of vertex-disjoint cycles C such that 9|I| ≥ 4n − 3|C|.

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Towards optimal kernel for connected vertex cover in planar graphs

Cited by 8 publications

References 23 publications

A $$14k$$ -Kernel for Planar Feedback Vertex Set via Region Decomposition

A $$14k$$ -Kernel for Planar Feedback Vertex Set via Region Decomposition

Nonblocker in H-minor free graphs: kernelization meets discharging

A 9k Kernel for Nonseparating Independent Set in Planar Graphs

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