Approximating the Edge Length of 2-Edge Connected Planar Geometric Graphs on a Set of Points

Dobrev, Štefan; Kranakis, Evangelos; Kriz̧anc, Danny; Morales-Ponce, Oscar; Stacho, Ladislav

doi:10.1007/978-3-642-29344-3_22

Cited by 4 publications

(4 citation statements)

References 15 publications

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“…There has been extensive work on augmenting disconnected plane graphs to connected ones (see [22] for a recent survey) or achieving good connectivity properties [1][2][3]12,24,25,27]. For abstract graphs, this corresponds to the classical connectivity augmentation problem in combinatorial optimization and has a rich history, as well.…”

Section: Introductionmentioning

confidence: 98%

“…A closely related family of problems is the graph augmentation, in which one would like to add new edges, ideally as few as possible, to a given graph in such a way that some desired property is achieved. There has been extensive work on augmenting disconnected plane graphs to connected ones (see [21] for a recent survey) or achieving good connectivity properties [1,2,3,12,23,24,26]. For abstract graphs, this corresponds to the classical connectivity augmentation problem in combinatorial optimization and has a rich history, as well.…”

Section: Introductionmentioning

confidence: 99%

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Geometric Biplane Graphs II: Graph Augmentation

García

Hurtado²,

Korman

et al. 2015

Graphs and Combinatorics

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We study biplane graphs drawn on a finite point set $S$ in the plane in general position. This is the family of geometric graphs whose vertex set is $S$ and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Geometric Biplane Graphs II: Graph Augmentation

García

Hurtado²,

Korman

et al. 2015

Graphs and Combinatorics

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“…The length of the edges in planar connectivity augmentation was studied only recently in the context of wireless networks. Given a PLSG G = (V, E) where the vertices induce a 2-edgeconnected unit disk graph, Dobrev et al [6] compute a 2-edge-connected PSLG by adding edges of length at most 2. Kranakis et al [16] studied the combined problem of adding the minimum number of edges of bounded length: A 2-edge-connected augmentation is possible such that |E + | is at most the number of bridges in G and max e ∈E + e ≤ 3 max e∈E e .…”

Section: Introductionmentioning

confidence: 99%

Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs

Akitaya

Inkulu

Nichols

et al. 2017

WALCOM: Algorithms and Computation

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We consider edge insertion and deletion operations that increase the connectivity of a given planar straight-line graph (PSLG), while minimizing the total edge length of the output. We show that every connected PSLG G = (V, E) in general position can be augmented to a 2connected PSLG (V, E ∪ E + ) by adding new edges of total Euclidean length E + ≤ 2 E , and this bound is the best possible. An optimal edge set E + can be computed in O(|V | 4 ) time; however the problem becomes NP-hard when G is disconnected. Further, there is a sequence of edge insertions and deletions that transforms a connected PSLG G = (V, E) into a planar straight-line cycle G = (V, E ) such that E ≤ 2 MST(V ) , and the graph remains connected with edge length below E + MST(V ) at all stages. These bounds are the best possible. IntroductionConnectivity augmentation is a classical problem in combinatorial optimization. Given a graph G = (V, E) and a parameter τ ∈ N, add a set of new edges E + of minimum cardinality or weight such that the augmented graph G = (V, E ∪ E + ) is τ -connected (resp., τ -edge-connected). Efficient algorithms are known for both connectivity and edge-connectivity augmentation over abstract graphs and constant τ [8,19,24]. In this paper we consider weighted connectivity augmentation for planar straight-line graphs (PSLGs). The vertices are points in Euclidean plane, the edges are noncrossing line segments between the corresponding vertices, and the weight of an edge is its Euclidean length.The edge-and node-connectivity of a planar graph is at most 5 by Euler's theorem. Further, not every PSLG can be augmented to a 3-connected (resp., 4-edge-connected) PSLG; see [12] for feasibility conditions. Finding the minimum number of edges to augment a given PSLG to τconnectivity or τ -edge-connectivity is NP-complete [22] for 2 ≤ τ ≤ 5; the reduction requires the input graph G to be disconnected (the NP-hardness claim for connected input [22, Corollary 2] turned out to be flawed). Worst case bounds are known for the most important cases: Every PSLG G with n vertices can be augmented to 2-edge-connectivity with at most (4n − 4)/3 edges [2];

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“…We focus on 2-edge connected planar graph. The results presented in Chapter 8 appear in the paper "Approximating the Edge Length of 2-Edge Connected Planar Geometric Graphs in UDGs" [18].…”

Section: Optimal Length ^-Connected Planar Graph Problemmentioning

confidence: 99%

Connectivity of wireless sensor networks using directional antennae

Ponce¹

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The author has granted a nonexclusive license allowing Library and Archives Canada to reproduce, publish, archive, preserve, conserve, communicate to the public by telecommunication or on the Internet, loan, distribute and sell theses worldwide, for commercial or noncommercial purposes, in microform, paper, electronic and/or any other formats. L'auteur a accorde une licence non exclusive permettant a la Bibliotheque et Archives Canada de reproduce, publier, archiver, sauvegarder, conserver, transmettre au public par telecommunication ou par I'lnternet, preter, distribuer et vendre des theses partout dans le monde, a des fins commerciales ou autres, sur support microforme, papier, electronique et/ou autres formats.

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Approximating the Edge Length of 2-Edge Connected Planar Geometric Graphs on a Set of Points

Cited by 4 publications

References 15 publications

Geometric Biplane Graphs II: Graph Augmentation

Geometric Biplane Graphs II: Graph Augmentation

Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs

Connectivity of wireless sensor networks using directional antennae

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