Spanning spiders and light-splitting switches

Gargano, Luisa; Hammar, Mikael; Hell, Pavol; Stacho, Ladislav; Vaccaro, Ugo

doi:10.1016/j.disc.2004.04.005

Cited by 45 publications

(32 citation statements)

References 15 publications

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“…The result subsumes known conditions for the traceability of such a graph G from [10,11]: Theorem 1 (Gargano et al [7]). If a graph G with no induced K 1,3 satisfies k+3 (G) n − k − 2, then G admits a spanning tree with at most k branching vertices.…”

Section: Introductionsupporting

confidence: 65%

“…Theorem 1 shows that k+2 may be replaced with k+3 for K 1,3 -free graphs, and an example in [7] proves that the bound in the theorem does not hold for graphs that may contain induced K 1,3 . On the other hand, the following possibility does not seem to be ruled out.…”

Section: Problem 8 Is It True That If G Is a -Connected Graph Andmentioning

confidence: 92%

“…Following [7], we call a spanning tree with at most one branching vertex a spanning spider and remark that the investigation of this sort of spanning trees was catalyzed by problems in the construction of optical networks. Yet another related constraint on spanning trees is an 1 Research supported by project 1M0545 and Research Plan MSM 4977751301 of the Czech Ministry of Education.…”

Section: Introductionmentioning

confidence: 99%

“…Gargano et al [7] prove a sufficient condition for a graph G without an induced K 1,3 to admit a spanning tree with a bounded number of branching vertices. The result subsumes known conditions for the traceability of such a graph G from [10,11]: Theorem 1 (Gargano et al [7]).…”

Section: Introductionmentioning

confidence: 99%

“…Sufficient conditions for the existence of extremal spanning trees of the above types have been studied e.g. in [1,7,8,12].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Neighborhood unions and extremal spanning trees

Flandrin¹,

Kaiser²,

Kuel³

et al. 2008

Discrete Mathematics

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Section: Introductionsupporting

confidence: 65%

Section: Problem 8 Is It True That If G Is a -Connected Graph Andmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Sufficient conditions for the existence of extremal spanning trees of the above types have been studied e.g. in [1,7,8,12].…”

Section: Introductionmentioning

confidence: 99%

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Neighborhood unions and extremal spanning trees

Flandrin¹,

Kaiser²,

Kuel³

et al. 2008

Discrete Mathematics

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Degree Conditions for Spanning Brooms

Chen

Ferrara²,

et al. 2014

Journal of Graph Theory

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A broom is a tree obtained by subdividing one edge of the star an arbitrary number of times. In (E. Flandrin, T. Kaiser, R. Kužel, H. Li and Z. Ryjáček, Neighborhood Unions and Extremal Spanning Trees, Discrete Math 308 (2008), 2343–2350) Flandrin et al. posed the problem of determining degree conditions that ensure a connected graph G contains a spanning tree that is a broom. In this article, we give one solution to this problem by demonstrating that if G is a connected graph of order n≥56 with δ(G)≥n−23, then G contains a spanning broom. This result is best possible.

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Leaf‐Critical and Leaf‐Stable Graphs

Wiener

2016

Journal of Graph Theory

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Abstract:The minimum leaf number ml(G) of a connected graph G is defined as the minimum number of leaves of the spanning trees of G if G is not hamiltonian and 1 if G is hamiltonian. We study nonhamiltonian graphs with the property l :. These graphs will be called (l + 1)-leaf-critical and l -leaf-stable, respectively. It is far from obvious whether such graphs exist; for example, the existence of 3-leaf-critical graphs (that turn out to be the so-called hypotraceable graphs) was an open problem until 1975. We show that l -leaf-stable and l -leaf-critical graphs exist for every integer l ≥ 2, moreover for n sufficiently large, planar l -leafstable and l -leaf-critical graphs exist on n vertices. We also characterize 2-fragments of leaf-critical graphs generalizing a lemma of Thomassen. As an application of some of the leaf-critical graphs constructed, we settle an open problem of Gargano et al. concerning spanning spiders. We also explore connections with a family of graphs introduced by Grünbaum in correspondence with the problem of finding graphs without concurrent longest paths.

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Spanning spiders and light-splitting switches

Cited by 45 publications

References 15 publications

Neighborhood unions and extremal spanning trees

Neighborhood unions and extremal spanning trees

Degree Conditions for Spanning Brooms

Leaf‐Critical and Leaf‐Stable Graphs

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