On the Hardness of Eliminating Small Induced Subgraphs by Contracting Edges

Lokshtanov, Daniel; Misra, Neeldhara; Saurabh, Saket

doi:10.1007/978-3-319-03898-8_21

Cited by 21 publications

(24 citation statements)

References 7 publications

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“…Therefore, the Clique Contraction problem can be seen as the parametric dual of the Hadwiger Number problem, and is NP-complete on general graphs. When parameterized by k, the Clique Contraction problem was recently shown to be fixed-parameter tractable [4,16], but the problem does not admit a polynomial kernel unless NP ⊆ coNP/ poly [4].…”

Section: Introductionmentioning

confidence: 99%

Hadwiger Number of Graphs with Small Chordality

Golovach¹,

Heggernes²,

Hof³

et al. 2015

SIAM J. Discrete Math.

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The Hadwiger number of a graph G is the largest integer h such that G has the complete graph K h as a minor. We show that the problem of determining the Hadwiger number of a graph is NP-hard on co-bipartite graphs, but can be solved in polynomial time on cographs and on bipartite permutation graphs. We also consider a natural generalization of this problem that asks for the largest integer h such that G has a minor with h vertices and diameter at most s. We show that this problem can be solved in polynomial time on AT-free graphs when s ≥ 2, but is NP-hard on chordal graphs for every fixed s ≥ 2.

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Section: Introductionmentioning

confidence: 99%

Hadwiger Number of Graphs with Small Chordality

Golovach¹,

Heggernes²,

Hof³

et al. 2015

SIAM J. Discrete Math.

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“…For example, when H is the class of chordal graphs, H-Vertex Deletion is fixed-parameter tractable when parameterized by k [25], contrasting the aforementioned W [2]-hardness result for the edge contraction variant [7,24]. Also, when H is the class of forests, H-Vertex Deletion admits a polynomial kernel with at most 4k 2 vertices [34], whereas the edge contraction variant does not admit a polynomial kernel, unless NP ⊆ coNP/poly [19].…”

Section: Introductionmentioning

confidence: 99%

“…The parameterized study of graph modification problems with respect to this operation has only recently been initiated, but has already proved to be very fruitful [7,8,[16][17][18][19][20][21]24]. In general, for every graph class H, the H-Contraction problem takes as input a graph G and an integer k, and asks whether there exists a graph H ∈ H such that G is k-contractible to H, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…A general result by Asano and Hirata [1] shows that this problem is NP-complete for many natural graph classes H. On the positive side, when parameterized by k, the problem is known to be fixed-parameter tractable when H is the class of paths [19], trees [19], bipartite graphs [18,21,26], planar graphs [16], or split graphs [8]. Very recently, two groups of authors independently showed that H-Contraction is W [2]-hard with respect to the same parameter when H is the class of chordal graphs [7,24], whereas the problem is still open for interval graphs. Interestingly, the problem admits a linear vertex kernel when H is the class of paths, but does not admit a polynomial kernel when H is the class of trees, unless NP ⊆ coNP/poly [19].…”

Section: Introductionmentioning

confidence: 99%

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Parameterized complexity of three edge contraction problems with degree constraints

et al. 2014

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2014) 'Parameterized complexity of three edge contraction problems with degree constraints.', Acta Informatica., 51 (7). pp. 473-497. Further information on publisher's website:http://dx.doi.org/10.1007/s00236-014-0204-z Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/s00236-014-0204-z Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. For any graph class H, the H-Contraction problem takes as input a graph G and an integer k, and asks whether there exists a graph H ∈ H such that G can be modified into H using at most k edge contractions. We study the parameterized complexity of H-Contraction for three different classes H: the class H ≤d of graphs with maximum degree at most d, the class H =d of dregular graphs, and the class of d-degenerate graphs. We completely classify the parameterized complexity of all three problems with respect to the parameters k, d, and d+k. Moreover, we show that H-Contraction admits an O(k) vertex kernel on connected graphs when H ∈ {H ≤2 , H=2}, while the problem is W[2]-hard when H is the class of 2-degenerate graphs and hence is expected not to admit a kernel at all. In particular, our results imply that H-Contraction admits a linear vertex kernel when H is the class of cycles.

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“…On the other hand P-Contraction admits a polynomial kernel with at most 5k + 3 vertices (see [18] for an improved bound of 3k + 4 on the number of vertices). Moreover, F-Contraction is not FPT(unless some unlikely collapse in Parameterized Complexity happens) even for simple family of graphs such as P t -free graphs for some t ≥ 5, the family of C t -free graphs for some t ≥ 4 [6,19], and the family of split graphs [2]. Here, P t and C t denotes the path and cycle on t vertices.…”

Section: Introductionmentioning

confidence: 99%

On the Parameterized Complexity of Contraction to Generalization of Trees

2018

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For a family of graphs F, the F-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists S ⊆ E(G) of size at most k such that G/S belongs to F. Here, G/S is the graph obtained from G by contracting all the edges in S. Heggernes et al. [Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied F-Contraction when F is a simple family of graphs such as trees and paths. In this paper, we study the F-Contraction problem, where F generalizes the family of trees. In particular, we define this generalization in a "parameterized way". Let T be the family of graphs such that each graph in T can be made into a tree by deleting at most edges. Thus, the problem we study is T -Contraction. We design an FPT algorithm for T -Contraction running in time O((2 √ + 2) O(k+ ) · n O(1) ). Furthermore, we show that the problem does not admit a polynomial kernel when parameterized by k. Inspired by the negative result for the kernelization, we design a lossy kernel for T -Contraction of size O([k(k + 2 )] ( α α−1 +1) ).

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On the Hardness of Eliminating Small Induced Subgraphs by Contracting Edges

Cited by 21 publications

References 7 publications

Hadwiger Number of Graphs with Small Chordality

Hadwiger Number of Graphs with Small Chordality

Parameterized complexity of three edge contraction problems with degree constraints

On the Parameterized Complexity of Contraction to Generalization of Trees

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