Certifying 3-Edge-Connectivity

Mehlhorn, Kurt; Neumann, Adrian; Schmidt, Jens M.

doi:10.1007/s00453-015-0075-x

Cited by 11 publications

(13 citation statements)

References 26 publications

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“…Find all subgraphs of G which are 3-edge-connected and contract each of them to a vertex; for example by choosing l ∈ L G such that the corresponding entries vanish and considering Γ(G; l). This yields a cactus graph with the property mentioned above [26]. The cactus graph has a tree-like structure.…”

Section: Supremizers Of Tree Graphs (Section 4)mentioning

confidence: 94%

Quantum Graphs which Optimize the Spectral Gap

Band

Lévy

2017

Ann. Henri Poincaré

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A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of the spectral gap (the first positive Laplacian eigenvalue) on the choice of edge lengths. In particular, starting from a certain discrete graph, we seek the quantum graph for which an optimal (either maximal or minimal) spectral gap is obtained. We fully solve the minimization problem for all graphs. We develop tools for investigating the maximization problem and solve it for some families of graphs.2000 Mathematics Subject Classification. 05C45, 34L15, 35Pxx, 35R02.

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Section: Supremizers Of Tree Graphs (Section 4)mentioning

confidence: 94%

Quantum Graphs which Optimize the Spectral Gap

Band

Lévy

2017

Ann. Henri Poincaré

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“…Our algorithm enumerates all the 3-edge-connected spanning subgraphs of a given plane graph with n vertices in O(n 2 ) time for each. On checking 3-edge-connectivity, although it is the same time complexity of the algorithm by Mehlhorn et al [5], our algorithm is extremely simple, which is also an advantage.…”

Section: Resultsmentioning

confidence: 99%

“…Here, we focus on the case of k = 3. In 2017, Mehlhorn et al proposed an algorithm for checking 3-edgeconnectivity of an input graph [5]. The time complexity of their algorithm is a linear time.…”

Section: Lemmamentioning

confidence: 99%

“…On the other hand, this paper also gives an algorithm that generates each 3-edge-connected spanning subgraph of an input plane graph in O(n 2 ) time by using an algorithm which can check 3-edge-connectivity in O(n) time for a given plane graph. In 2017, for a given graph, Mehlhorn et al already proposed an algorithm for checking 3-edge-connectivity in O(n) time [5]. So, both algorithms require the same time complexity.…”

Section: Introductionmentioning

confidence: 99%

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A Note on Enumeration of 3-Edge-Connected Spanning Subgraphs in Plane Graphs

Matsui

Ozeki

2021

IEICE Trans. Inf. & Syst.

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This paper deals with the problem of enumerating 3-edgeconnected spanning subgraphs of an input plane graph. In 2018, Yamanaka et al. proposed two enumeration algorithms for such a problem. Their algorithm generates each 2-edge-connected spanning subgraph of a given plane graph with n vertices in O(n) time, and another one generates each k-edgeconnected spanning subgraph of a general graph with m edges in O(mT) time, where T is the running time to check the k-edge connectivity of a graph. This paper focuses on the case of the 3-edge-connectivity in a plane graph. We give an algorithm which generates each 3-edge-connected spanning subgraph of the input plane graph in O(n 2) time. This time complexity is the same as the algorithm by Yamanaka et al., but our algorithm is simpler than theirs.

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“…We give a conceptually very simple planarity test based on Mondshein's sequence for any 3connected graph G in time O(n). The 3-connectivity requirement is not crucial, as the planarity of G can be reduced to the planarity of all 3-connected components of G, which in turn are computed as a side-product from the computation of the BG-sequence [28,Appendix 2]. Alternatively, one could also use standard algorithms [18,16] for reducing G to be 3-connected.…”

Section: Application 3: Planarity Testingmentioning

confidence: 99%

Mondshein Sequences (a.k.a. (2,1)-Orders)

Schmidt¹

2016

SIAM J. Comput.

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Canonical orderings [STOC'88, FOCS'92] have been used as a key tool in graph drawing, graph encoding and visibility representations for the last decades. We study a far-reaching generalization of canonical orderings to non-planar graphs that was published by Lee Mondshein in a PhD-thesis at M.I.T. as early as 1971.Mondshein proposed to order the vertices of a graph in a sequence such that, for any i, the vertices from 1 to i induce essentially a 2-connected graph while the remaining vertices from i + 1 to n induce a connected graph. Mondshein's sequence generalizes canonical orderings and became later and independently known under the name non-separating ear decomposition. Surprisingly, this fundamental link between canonical orderings and non-separating ear decomposition has not been established before. Currently, the fastest known algorithm for computing a Mondshein sequence achieves a running time of O(nm); the main open problem in Mondshein's and follow-up work is to improve this running time to subquadratic time.After putting Mondshein's work into context, we present an algorithm that computes a Mondshein sequence in optimal time and space O(m). This improves the previous best running time by a factor of n. We illustrate the impact of this result by deducing linear-time algorithms for five other problems, for four out of which the previous best running times have been quadratic. In particular, we show how to -compute three independent spanning trees in a 3-connected graph in time O(m), improving a result of Cheriyan and Maheshwari [J. Algorithms 9(4)], -improve the preprocessing time from O(n 2 ) to O(m) for the output-sensitive data structure by Di Battista, Tamassia and Vismara [Algorithmica 23(4)] that reports three internally disjoint paths between any given vertex pair, -derive a very simple O(n)-time planarity test once a Mondshein sequence has been computed, -compute a nested family of contractible subgraphs of 3-connected graphs in time O(m), -compute a 3-partition in time O(m), while the previous best running time is O(n 2 ) due to Suzuki et al. [IPSJ 31(5)].

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Certifying 3-Edge-Connectivity

Cited by 11 publications

References 26 publications

Quantum Graphs which Optimize the Spectral Gap

Quantum Graphs which Optimize the Spectral Gap

A Note on Enumeration of 3-Edge-Connected Spanning Subgraphs in Plane Graphs

Mondshein Sequences (a.k.a. (2,1)-Orders)

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