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

Golovach, Petr A.; Heggernes, Pinar; Kratsch, Dieter; Lima, Paloma T.; Paulusma, Daniël

doi:10.1007/978-3-319-68705-6_21

Cited by 3 publications

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“…Determining the complexity of H-SQUARE ROOT when H is the class of planar graphs is still a wide open problem. Golovach et al [12] also proved that squares of graphs of pathwidth at most 2 can be recognized in polynomial time. We recall that every cactus has treewidth at most 2.…”

Section: Discussionmentioning

confidence: 99%

“…In fact, our algorithm can be modified to find a cactus root in the same time (if it exists). Every cactus is outerplanar, and recently Golovach et al [12] proved that squares of outerplanar graphs can be recognized in polynomial time. Determining the complexity of H-SQUARE ROOT when H is the class of planar graphs is still a wide open problem.…”

Section: Discussionmentioning

confidence: 99%

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Finding Cactus Roots in Polynomial Time

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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%

Section: Discussionmentioning

confidence: 99%

Finding Cactus Roots in Polynomial Time

et al. 2017

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“…, B 4 . This implies that in order to prove condition (ii) it suffices to find a vertex v i that appears in at least five bags of B ∪ B 16 .…”

Section: Structural Lemmasmentioning

confidence: 99%

“…We may assume without loss of generality that B 1 = {x 1 , u, v 1 } and B 16 = {x 16 , u, v 2 }. Recall that v 1 does not belong to any bag B i for i ≥ 5 and x 16 belongs to B 16 , while u belongs to any bag between B 1 and B 16 . Then there exists a bag {v 1 , v 2 , u}, which has to be between B 1 = {x 1 , u, v 1 } and B 16 = {x 16 , u, v 2 }.…”

Section: Structural Lemmasmentioning

confidence: 99%

“…By the above assumption, v does not belong to B i for i ≥ 5. In particular, this means that v is not adjacent to x 16 . As x 1 and x 16 are true twins in G, we find that v is a neighbour of x 16 in G. As v and x 16 are not adjacent in H, this means that G contains a path vwx 16 for some vertex w. If w = u, then H contains a path x 1 vwx 16 with u / ∈ {v, w}.…”

Section: Structural Lemmasmentioning

confidence: 99%

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

Golovach

Heggernes

Kratsch

et al. 2017

Graph-Theoretic Concepts in Computer Science

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Abstract. Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomialtime algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2.

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

Cited by 3 publications

References 32 publications

Finding Cactus Roots in Polynomial Time

Finding Cactus Roots in Polynomial Time

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

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

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