Space graphs and sphericity

Maehara, Hiroshi

doi:10.1016/0166-218x(84)90113-6

Cited by 44 publications

(17 citation statements)

References 5 publications

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“…The sphericity sph(G) of a graph G is the least k such that G is a k-sphere graph. (Every graph has finite sphericity [11].) Sphericity was first introduced by Havel [7] in his PhD thesis to study the geometry of molecules.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Sphere and dot product representations of graphs

Kang

Müller

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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A graph G is a k-sphere graph if there are k-dimensional real vectors v1, . . . , vn such that ij ∈ E(G) if and only if the distance between vi and vj is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v1, . . . , vn such that ij ∈ E(G) if and only if the dot product of vi and vj is at least 1.By relating these two geometric graph constructions to oriented k-hyperplane arrangements, we prove that the problems of deciding, given a graph G, whether G is a k-sphere or a k-dot product graph are NP-hard for all k > 1. In the former case, this proves a conjecture of Breu and Kirkpatrick [2]. In the latter, this answers a question of Fiduccia, Scheinerman, Trenk and Zito [5].Furthermore, motivated by the question of whether these two recognition problems are in NP, as well as by the implicit graph conjecture, we demonstrate that, for all k > 1, there exist k-sphere graphs and k-dot product graphs such that each representation in k-dimensional real vectors needs at least an exponential number of bits to be stored in the memory of a computer. On the other hand, we show that exponentially many bits are always enough. This resolves a question of Spinrad [22].

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Sphere and dot product representations of graphs

Kang

Müller

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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show abstract

“…The sphericity sph(G) of a graph G is the least k such that G is a k-sphere graph. (Every graph has finite sphericity [10].) Sphericity was first introduced by Havel [7] in his Ph.D. thesis to study the geometry of molecules.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Sphere and Dot Product Representations of Graphs

Kang

Müller

2012

Discrete Comput Geom

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A graph G is a k-sphere graph if there are k-dimensional real vectors v 1 , . . . , v n such that ij ∈ E(G) if and only if the distance between v i and v j is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v 1 , . . . , v n such that ij ∈ E(G) if and only if the dot product of v i and v j is at least 1.By relating these two geometric graph constructions to oriented k-hyperplane arrangements, we prove that the problems of deciding, given a graph G, whether G is a k-sphere or a k-dot product graph are NP-hard for all k > 1. In the former case, this proves a conjecture of Breu and Kirkpatrick (Comput. Geom. 9:3-24, 1998). In the latter, this answers a question of Fiduccia et al. (Discrete Math. 181:113-138, 1998).Furthermore, motivated by the question of whether these two recognition problems are in NP, as well as by the implicit graph conjecture, we demonstrate that, for all k > 1, there exist k-sphere graphs and k-dot product graphs such that each representation in k-dimensional real vectors needs at least an exponential number of bits to be stored in the memory of a computer. On the other hand, we show that exponentially many bits are always enough. This resolves a question of Spinrad (Efficient Graph Representations, 2003).

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“…They also pointed out that when d = 2, there are point graphs that are not planar and there are planar graphs that are not point graphs. Similar studies were conducted in the area of discrete mathematics, where symmetric point graphs are called space graphs [3] and symmetric planar point graphs are called unit disk graphs [2].…”

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confidence: 92%

Topological Characteristics of Random Multihop Wireless Networks

2005

Cluster Comput

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Multihop wireless networks are treated as random symmetric planar point graphs, where all the nodes have the same transmission power and radius, and vertices of a graph are drawn randomly over certain geographical region. Several basic and important topological properties of random multihop wireless networks are studied, including node degree, connectivity, diameter, bisection width, and biconnectivity. It is believed that such study has very useful implication in real applications.

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Space graphs and sphericity

Cited by 44 publications

References 5 publications

Sphere and dot product representations of graphs

Sphere and dot product representations of graphs

Sphere and Dot Product Representations of Graphs

Topological Characteristics of Random Multihop Wireless Networks

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