Dirac's Condition for Completely Independent Spanning Trees

Araki, Tomoyuki

doi:10.1002/jgt.21780

Cited by 53 publications

(26 citation statements)

References 5 publications

(10 reference statements)

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“…These conditions are based on the sufficient conditions for hamiltonicity: Dirac's condition [25] and Ore's condition [26]. The Dirac's condition has been generalized to more than two trees [27]- [29] and has been independently improved [28], [29] for two trees.…”

Section: Related Workmentioning

confidence: 99%

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Completely independent spanning trees for enhancing the robustness in ad-hoc Networks

Moinet¹,

Darties²,

Gastineau³

et al. 2017

2017 IEEE 13th International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob)

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Abstract-We investigate the problem of computing Completely Independent Spanning Trees (CIST) under a practical approach. We aim to show that despite CISTs are very challenging to exhibit in some networks, they present a real interest in ad-hoc networks and can be computed to enhance the network robustness. We propose an original ILP formulation for CISTs and we show through simulation results on representative network models that several CISTs can be computed when the network density is sufficiently high. These results tend to reinforce the interest of CISTs for various network operations such as robustness, load-balancing, traffic splitting, . . . As an important point, our results show that both the density and the number of nodes have an impact on the number of CISTs that can be found on ad-hoc networks.

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Section: Related Workmentioning

confidence: 99%

“…We placed 50 nodes uniformly, then we searched for (a lower bound on) the number of CISTs and its upper bound, for transmission range values r in [20,25,30,35,40,45]. We repeated the measurements for 10 different simulated networks.…”

Section: Impact Of the Transmission Rangementioning

confidence: 99%

Completely independent spanning trees for enhancing the robustness in ad-hoc Networks

Moinet¹,

Darties²,

Gastineau³

et al. 2017

2017 IEEE 13th International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob)

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show abstract

Section: Introductionmentioning

confidence: 99%

“…Hasunuma [8] conjectured that there are k CISTs in any 2k-connected graph and showed the NP-completeness for determining the existence of two CISTs in an arbitrary graph G. So far the study related to CISTs has received less attention except for [1], [8]- [10]. Araki [1] showed that a graph G of n vertices has two CISTs if the minimum degree of G is at least n/2, and the square of a 2-connected graph has two CISTs. Hasunuma showed that there are k CISTs in the underlying graph of any k-connected line digraph L(G) [8], there are two CISTs in any 4-connected maximal planar graph [9], and there are two CISTs in the Cartesian product of any 2-connected graphs [10].…”

Section: Introductionmentioning

confidence: 99%

Completely Independent Spanning Trees on Some Interconnection Networks

Pai

Yang

Yao

et al. 2014

IEICE Trans. Inf. & Syst.

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SUMMARYLet T 1 , T 2 , . . . , T k be spanning trees in a graph G. If, for any two vertices u, v of G, the paths joining u and v on the k trees are mutually vertex-disjoint, then T 1 , T 2 , . . . , T k are called completely independent spanning trees (CISTs for short) of G. The construction of CISTs can be applied in fault-tolerant broadcasting and secure message distribution on interconnection networks. Hasunuma (2001) first introduced the concept of CISTs and conjectured that there are k CISTs in any 2k-connected graph. Unfortunately, this conjecture was disproved by Péterfalvi recently. In this note, we give a necessary condition for k-connected k-regular graphs with k/2 CISTs. Based on this condition, we provide more counterexamples for Hasunuma's conjecture. By contrast, we show that there are two CISTs in 4-regular chordal rings CR (N, d) with N = k(d − 1) + j under the condition that k 4 is even and 0 j 4. In particular, the diameter of each constructed CIST is derived.

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“…In [4], Hasunuma study the completely independent spanning trees in some graphs. Recently, Araki [1] gave a Dirac's condition characterization for the existence of completely independent spanning trees. Let G be a graph and V 1 , V 2 be two disjoint sets of vertices in G. Denote by B(V 1 , V 2 ) the bipartite graph induced by the edges with one end in V 1 and the other end in V 2 .…”

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confidence: 99%

Ore’s condition for completely independent spanning trees

Fan

Hong

Liu

2014

Discrete Applied Mathematics

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a b s t r a c tTwo edge-disjoint spanning trees of a graph G are completely independent if the two paths connecting any two vertices of G in the two trees are internally disjoint. It has been asked whether sufficient conditions for hamiltonian graphs are also sufficient for the existence of two completely independent spanning trees (CISTs). We prove that it is true for the classical Ore-condition. That is, if G is a graph on n vertices in which each pair of non-adjacent vertices have degree-sum at least n, then G has two CISTs. It is known that the line graph of every 4-edge connected graph is hamiltonian. We prove that this is also true for CISTs: the line graph of every 4-edge connected graph has two CISTs. Thomassen conjectured that every 4-connected line graph is hamiltonian. Unfortunately, being 4-connected is not enough for the existence of two CISTs in line graphs. We prove that there are infinitely many 4-connected line graphs that do not have two CISTs.

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Dirac's Condition for Completely Independent Spanning Trees

Cited by 53 publications

References 5 publications

Completely independent spanning trees for enhancing the robustness in ad-hoc Networks

Completely independent spanning trees for enhancing the robustness in ad-hoc Networks

Completely Independent Spanning Trees on Some Interconnection Networks

Ore’s condition for completely independent spanning trees

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