A linear kernel for finding square roots of almost planar graphs

Golovach, Petr A.; Kratsch, Dieter; Paulusma, Daniël; Stewart, Anthony

doi:10.1016/j.tcs.2017.05.008

Cited by 11 publications

(24 citation statements)

References 27 publications

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“…This leads to the following (known) generalization: a cactus block graph is a connected graph, in which each block is a cycle or a complete graph. Recently, [14] polynomial Line graphs [24] polynomial Trivially perfect graphs [25] polynomial…”

Section: Discussionmentioning

confidence: 99%

Finding Cactus Roots in Polynomial Time

et al. 2017

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A graph H is a square root of a graph G, or equivalently, G is the square of H , if G can be obtained from H by adding an edge between any two vertices in H that are of distance 2. The SQUARE ROOT problem is that of deciding whether a given graph admits a square root. The problem of testing whether a graph admits a square root which belongs to some specified graph class H is called the H-SQUARE ROOT problem. By showing boundedness of treewidth we prove that SQUARE ROOT is polynomial-time solvable on some classes of graphs with small clique number and that H-SQUARE ROOT is polynomial-time solvable when H is the class of cactuses.

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Section: Discussionmentioning

confidence: 99%

Finding Cactus Roots in Polynomial Time

et al. 2017

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“…The last column of this table indicates whether the squares of the graph class have bounded treewidth, where an ⇤ means that these squares have bounded treewidth after some appropriate edge reduction (see [3] for further details). Note that the seven graph classes in the bottom seven rows not only have bounded treewidth but also have bounded clique number.…”

Section: Lemma 1 ([2])mentioning

confidence: 99%

Squares of Low Clique Number

Golovach

Kratsch

Paulusma

et al. 2016

Electronic Notes in Discrete Mathematics

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Citation for published item:qolovhD etr nd urtshD hieter nd ulusmD hni¤ el nd tewrtD enthony @PHITA 9qures of low lique numerF9D iletroni notes in disrete mthemtisFD SS F ppF IWSEIWVF Further information on publisher's website: 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. AbstractThe Square Root problem is that of deciding whether a given graph admits a square root. This problem is only known to be NP-complete for chordal graphs and polynomial-time solvable for non-trivial minor-closed graph classes and a very limited number of other graph classes. By researching boundedness of the treewidth of a graph, we prove that Square Root is polynomial-time solvable on various graph classes of low clique number that are not minor-closed.

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“…On the negative side, Square Root is NP-complete on chordal graphs [23]. There also exist a number of parameterized complexity results for the problem [8,19].…”

Section: Introductionmentioning

confidence: 99%

“…The goal is to obtain a graph whose treewidth is bounded by a constant, which enables us to solve the problem in polynomial time after expressing it in Monadic Second-Order Logic and applying a classical result of Courcelle [9]. This idea has been used before (see, for instance, [7,8,18,19]), but in this paper we formalize the idea into a general framework. We discuss this framework in detail in Section 3.…”

Section: Introductionmentioning

confidence: 99%

“…On the negative side, Square Root is NP-complete on chordal graphs [23]. There also exist a number of parameterized complexity results for the problem [8,19].The intractability of Square Root has also been attacked by restricting properties of the square root. In this case, the input graph G is an arbitrary graph, and the question is whether G has a square root that belongs to some graph class H specified in advance.…”

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Algorithms for Outerplanar Graph Roots and Graph Roots of Pathwidth at Most 2

et al. 2019

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The 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. AbstractDeciding if a graph has a square root is a classical problem, which has been studied extensively both from graph-theoretic and algorithmic perspective. As the problem is NP-complete, substantial effort has been dedicated to determining the complexity of deciding if a graph has a square root belonging to some specific graph class H. There are both polynomialtime solvable and NP-complete results in this direction, depending on H.1 belongs to K j . Unfortunately, this characterization does not yield a polynomialtime algorithm for deciding whether G has a square root. This problem is called the Square Root problem. In 1994, Motwani and Sudan [31] proved that Square Root is NP-complete.Motivated by its computational hardness, special cases of Square Root have been studied where the input graph G belongs to a particular graph class. It is known that Square Root is polynomial-time solvable on planar graphs [28], and more generally, on every non-trivial minor-closed graph class [33]. Polynomial-time algorithms also exist if the input graph G belongs to one of the following graph classes: block graphs [26], line graphs [29], trivially perfect graphs [30], threshold graphs [30], graphs of maximum degree 6 [7], graphs of maximum average degree smaller than 46 11 [18] 1 graphs with clique number at most 3 [18], and graphs with bounded clique number and no long induced path [18]. On the negative side, Square Root is NP-complete on chordal graphs [23]. There also exist a number of parameterized complexity results for the problem [8,19].The intractability of Square Root has also been attacked by restricting properties of the square root. In this case, the input graph G is an arbitrary graph, and the question is whether G has a square root that belongs to some graph class H specified in advance. This problem is called H-Square Root, and this is the problem which we focus on in this paper.Significant advances have also been made on the complexity of H-Square Root. Previous results show that H-Square Root is polynomial-time solvable for the following graph classes H: trees [28], proper interval graphs [23], bipartite graphs [22], block graphs [26], strongly chordal split graphs [27], ptolemaic graphs [24], 3-sun-free split graphs [24], cactus graphs [18], cactus block graphs [12] and graphs with girth at least g for any fixed g ≥ 6 [14]. The result for 3-sun-free split graphs was extended to a number of other subclasses of split graphs in [25]. We observe that if H-Square Root is polynomial-time s...

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A linear kernel for finding square roots of almost planar graphs

Cited by 11 publications

References 27 publications

Finding Cactus Roots in Polynomial Time

Finding Cactus Roots in Polynomial Time

Squares of Low Clique Number

Algorithms for Outerplanar Graph Roots and Graph Roots of Pathwidth at Most 2

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