Dividing a Graph into Triconnected Components

Hopcroft, John E.; Tarjan, Robert E.

doi:10.1137/0202012

Cited by 742 publications

(550 citation statements)

References 7 publications

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“…So computing the k-tuple is the key step of the whole algorithm. But from Section 2., we know when k = 2, the complexity is O(|V | + |E|) [11]; when k = 3, the complexity is O(|V | 2 ) [18]. From Theorem 2.5, for the constraint graph in 2D, we know k ≤ 3; for the constraint graph in 3D k ≤ 7.…”

Section: If K ≥ |V (T )| − 2 the Algorithm Terminates; Else Goto Stepmentioning

confidence: 93%

“…Hopcroft and Tarjan gave an algorithm for testing vertex triconnectivity and for finding separating pairs in a bi-connected graph in O(|V | + |E|) time [11]. Kanevsky and Ramachandran offered an algorithm based on ear decomposition for testing 4-connectivity and for finding all separating triplets in a tri-connected graph in O(|V | 2 ) time [18].…”

Section: G Is Structurally Well-constrained If It Is Neither Structurmentioning

confidence: 99%

See 1 more Smart Citation

Geometric Constraint Solving Based on Connectivity of Graph

Zhang¹,

Gao²

2004

Computer-Aided Design and Applications

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Section: If K ≥ |V (T )| − 2 the Algorithm Terminates; Else Goto Stepmentioning

confidence: 93%

Section: G Is Structurally Well-constrained If It Is Neither Structurmentioning

confidence: 99%

Geometric Constraint Solving Based on Connectivity of Graph

Zhang¹,

Gao²

2004

Computer-Aided Design and Applications

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“…Various tests for 3-connectivity are known, and we refer the reader to [29], [37] for details including measures of the complexity of the tests involved. Tests for redundant rigidity in IR 2 have been derived [24] based on variants of Laman's theorem [35].…”

Section: Theorem 6 ( [30])mentioning

confidence: 99%

A Theory of Network Localization

Aspnes

Eren

Goldenberg

et al. 2006

IEEE Trans. on Mobile Comput.

563

396

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In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks.We further study the computational complexity of network localization and investigate a subclass of grounded graphs where localization can be computed efficiently. We conclude with a discussion of localization in sensor networks where the sensors are placed randomly.

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“…We use a standard tool for decomposing 2-connected graphs, the SPQR tree [5,6,16,17,19]. An SPQR tree for a graph G is a tree structure in which each tree node is associated with a graph C i , known as a 3-connected component of G. In each 3-connected component, some of the edges are labeled as "virtual", and each edge of the SPQR-tree is labeled by a pair of oriented virtual edges from its two endpoints; each virtual edge of a component is associated in this way with exactly one tree edge.…”

Section: Two-connected Subcubic Graphsmentioning

confidence: 99%

Planar Lombardi Drawings for Subcubic Graphs

Eppstein

2013

Graph Drawing

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Abstract. We prove that every planar graph with maximum degree three has a planar drawing in which the edges are drawn as circular arcs that meet at equal angles around every vertex. Our construction is based on the Koebe-Thurston-Andreev circle packing theorem, and uses a novel type of Voronoi diagram for circle packings that is invariant under Möbius transformations, defined using three-dimensional hyperbolic geometry. We also use circle packing to construct planar Lombardi drawings of a special class of 4-regular planar graphs, the medial graphs of polyhedral graphs, and we show that not every 4-regular planar graph has a Lombardi drawing. We have implemented our algorithm for 3-connected planar cubic graphs.

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Dividing a Graph into Triconnected Components

Cited by 742 publications

References 7 publications

Geometric Constraint Solving Based on Connectivity of Graph

Geometric Constraint Solving Based on Connectivity of Graph

A Theory of Network Localization

Planar Lombardi Drawings for Subcubic Graphs

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