Recognizing string graphs in NP

Schaefer, Marcus; Sedgwick, Eric; Štefankovič, Daniel

doi:10.1145/509909.509910

Cited by 53 publications

(77 citation statements)

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“…Garey and Johnson showed that the crossing number problem is NP-complete [14]. This proof also shows that pcr is NP-hard, but it does not imply that the problem lies in NP; that was established later in connection with the string graph problem [43]. The variant ocr is NP-complete as shown by Pach and Tóth [29]; NP-hardness is a modification of the Garey-Johnson proof and containment in NP relies on a refinement of Theorem 1.18 which takes into account the rotation system.…”

Section: Crossing Numbersmentioning

confidence: 95%

“…(This turns out to be highly non-trivial in the case of ocr.) All three problems lie in NP; for iocr this follows directly from Theorem 1.18, for pcr − and cr− the situation is more complicated, since there is no immediate bound on the number of crossings that do not count; using techniques from [43] and [44] the problems can be placed in NP (the upper bounds on the uncounted crossings are exponential). We do not know the complexity of iacr, though it is quite possible that the iocr-hardness proof can be adapted.…”

Section: Crossing Numbersmentioning

confidence: 99%

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Hanani-Tutte and Related Results

Schaefer

2013

Bolyai Society Mathematical Studies

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We investigate under what conditions crossings of adjacent edges and pairs of edges crossing an even number of times are unnecessary when drawing graphs. This leads us to explore the Hanani-Tutte theorem and its close relatives, emphasizing the intuitive geometric content of these results. The Hanani-Tutte Theorem in The PlaneIn 1934, Hanani [15] published a paper which-in passing-established the following result:Any drawing of a K 5 or a K 3,3 contains two independent edges crossing each other oddly. 1Since by Kuratowski's theorem every non-planar graph contains a subdivision of K 5 or K 3,3 , Hanani's observation implies that any drawing of a nonplanar graph contains two vertex-disjoint paths that cross an odd number of times and therefore two independent edges that cross oddly-one in each path. This consequence was first explicitly stated by Tutte [49]. 2 1 The result can be hard to find even if one reads German. It is stated as (1) on page 137 of the article and mainly an application of methods developed by Flores [20, 62. Kolloqium].2 The theorem is generally known as the Hanani-Tutte theorem, though Levow [17] calls it the "van Kampen-Shapiro-Wu characterization of planar graphs" emphasizing the parallel history of the theorem in algebraic topology (ignoring Flores, however). In a recent paper [34] we introduced the name "strong Hanani-Tutte theorem" to distinguish it from a weaker version that is also often called the Hanani-Tutte theorem in the literature.

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Section: Crossing Numbersmentioning

confidence: 95%

Section: Crossing Numbersmentioning

confidence: 99%

Hanani-Tutte and Related Results

Schaefer

2013

Bolyai Society Mathematical Studies

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“…The membership in N P follows from [11]. The N P-hardness is proved by means of a polynomial-time reduction from problem SUNFLOWER SEFE, which has been proved N P-complete for k = 3 graphs, even when the common graph is a tree T and the exclusive edges of each graph only connect leaves of the tree [2].…”

Section: Complexitymentioning

confidence: 99%

“…The membership in N P follows from [11]. In the following we describe a reduction that, given a 3-SAT formula ϕ, produces a streamed graph that is ω 0 -stream planar if and only if ϕ is satisfiable.…”

Section: Complexitymentioning

confidence: 99%

Planarity of Streamed Graphs

Lozzo

Rutter

2015

Lecture Notes in Computer Science

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Abstract. In this paper we introduce a notion of planarity for graphs that are presented in a streaming fashion. A streamed graph is a stream of edges e1, e2, . . . , em on a vertex set V . A streamed graph is ω-stream planar with respect to a positive integer window size ω if there exists a sequence of planar topological drawings Γi of the graphs Gi = (V, {ej | i ≤ j < i + ω}) such that the common graphis drawn the same in Γi and in Γi+1, for 1 ≤ i < m − ω. The STREAM PLANARITY Problem with window size ω asks whether a given streamed graph is ω-stream planar. We also consider a generalization, where there is an additional backbone graph whose edges have to be present during each time step. These problems are related to several well-studied planarity problems. We show that the STREAM PLANARITY Problem is N P-complete even when the window size is a constant and that the variant with a backbone graph is N P-complete for all ω ≥ 2. On the positive side, we provide O(n + ωm)-time algorithms for (i) the case ω = 1 and (ii) all values of ω provided the backbone graph consists of one 2-connected component plus isolated vertices and no stream edge connects two isolated vertices. Our results improve on the HananiTutte-style O((nm) 3 )-time algorithm proposed by Schaefer [GD'14] for ω = 1.

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“…For most of the intersection defined classes the complexity of their recognition is known. For example, interval graphs can be recognized in linear time [3], while recognition of string graphs is NP-complete [13,19]. Our goal is to study the easily recognizable classes and explore if the recognition problem becomes harder when extra conditions are given with the input.…”

Section: Introductionmentioning

confidence: 99%

Extending Partial Representations of Function Graphs and Permutation Graphs

Klavík

Kratochvíl

Krawczyk

et al. 2012

Lecture Notes in Computer Science

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Abstract. Function graphs are graphs representable by intersections of continuous real-valued functions on the interval [0,1] and are known to be exactly the complements of comparability graphs. As such they are recognizable in polynomial time. Function graphs generalize permutation graphs, which arise when all functions considered are linear. We focus on the problem of extending partial representations, which generalizes the recognition problem. We observe that for permutation graphs an easy extension of Golumbic's comparability graph recognition algorithm can be exploited. This approach fails for function graphs. Nevertheless, we present a polynomial-time algorithm for extending a partial representation of a graph by functions defined on the entire interval [0, 1] provided for some of the vertices. On the other hand, we show that if a partial representation consists of functions defined on subintervals of [0,1], then the problem of extending this representation to functions on the entire interval [0, 1] becomes NP-complete.

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Recognizing string graphs in NP

Cited by 53 publications

References 0 publications

Hanani-Tutte and Related Results

Hanani-Tutte and Related Results

Planarity of Streamed Graphs

Extending Partial Representations of Function Graphs and Permutation Graphs

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