Planar segment visibility graphs

Everett, Hazel; Hoàng, Chı́nh T.; Kilakos, K.; Noy, Marc

doi:10.1016/s0925-7721(00)00009-2

Cited by 8 publications

(12 citation statements)

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“…To our surprise, triangles seem to play a more fundamental role than other rigid structures, despite the close connections between rigidity and embedding with specified edge lengths [8,17,19]. Other than visibility graphs [9,15] and dense graphs [1], our results are the first positive results for efficient embeddings of (special) graphs with specified edge lengths.…”

Section: Introductionmentioning

confidence: 62%

“…Can we recognize realizable length assignments? These three problems have an extensive history, having been studied in the fields of computational geometry [9,15,27,29], rigidity theory [8,17,19], sensor networks [6,26], and structural analysis of molecules [1,10,18]. This reconstruction problem arises frequently when only distance information is known about a given structure, such as the atoms in a protein [1,10,18] or the nodes in an ad-hoc wireless network [6,25,26].…”

Section: Introductionmentioning

confidence: 99%

“…A reconstruction is always unique and easy-to-compute for a complete graph of (exact) distances, or any graph that can be "shelled" by incrementally locating nodes according to the distances to three noncollinear located neighbors ( Figure 1). More interesting is that visibility graphs [9] and segment visibility graphs [15] can be incrementally "shelled". In general, however, the reconstruction problem is NP-hard [29], even in the strong sense [27].…”

Section: Introductionmentioning

confidence: 99%

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Planar Embeddings of Graphs with Specified Edge Lengths

Cabello¹,

Demaine²,

Rote³

2007

JGAA

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We consider the problem of finding a planar straight-line embedding of a graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see that the problem is NP-hard. In contrast, we show that the problem is tractable-indeed, solvable in linear time on a real RAM-for straight-line embeddings of planar 3-connected triangulations, even if the outer face is not a triangle. This result is essentially tight: the problem becomes NP-hard if we consider instead straight-line embeddings of planar 3-connected infinitesimally rigid graphs with unit edge lengths, a natural relaxation of triangulations in this context.

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Section: Introductionmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Planar Embeddings of Graphs with Specified Edge Lengths

Cabello¹,

Demaine²,

Rote³

2007

JGAA

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“…Can we recognize realizable length assignments? These three problems have an extensive history, having been studied in the fields of computational geometry [8,14,26,27], rigidity theory [7,16,18], sensor networks [5,25], and structural analysis of molecules [1,9,17]. The reconstruction problem arises frequently when only distance information is known about a given structure, such as the atoms in a protein [1,9,17] or the nodes in an ad-hoc wireless network [5,24,25].…”

Section: Introductionmentioning

confidence: 99%

“…1). More interesting is that such graphs include visibility graphs [8] and segment visibility graphs [14]. In general, however, the reconstruction problem is NP-hard [27], even in the strong sense [26].…”

Section: Introductionmentioning

confidence: 99%

Planar Embeddings of Graphs with Specified Edge Lengths

2004

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Abstract.We consider the problem of finding a planar embedding of a (planar) graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see that the problem is NP-hard. In contrast, we show that the problem is tractable-indeed, solvable in linear time on a real RAM-for planar embeddings of planar 3-connected triangulations, even if the outer face is not a triangle. This result is essentially tight: the problem becomes NP-hard if we consider instead planar embeddings of planar 3-connected infinitesimally rigid graphs, a natural relaxation of triangulations in this context.

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Unit Disk Visibility Graphs

Ağaoğlu

Çağırıcı

2021

Trends in Mathematics

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We study unit disk visibility graphs, where the visibility relation between a pair of geometric entities is defined by not only obstacles, but also the distance between them. That is, two entities are not mutually visible if they are too far apart, regardless of having an obstacle between them. This particular graph class models real world scenarios more accurately compared to the conventional visibility graphs. We first define and classify the unit disk visibility graphs, and then show that the 3-coloring problem is NP-complete when unit disk visibility model is used for a set of line segments (which applies to a set of points) and for a polygon with holes.

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Planar segment visibility graphs

Cited by 8 publications

References 0 publications

Planar Embeddings of Graphs with Specified Edge Lengths

Planar Embeddings of Graphs with Specified Edge Lengths

Planar Embeddings of Graphs with Specified Edge Lengths

Unit Disk Visibility Graphs

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