Degree Conditions for Spanning Brooms

Chen, Guantao; Ferrara, Michael; Hu, Zhiquan; Jacobson, Michael S.; Liu, Huiqing

doi:10.1002/jgt.21784

Cited by 6 publications

(6 citation statements)

References 10 publications

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“…Further, the results in this article can be used to give conditions whether a graph contains a k-balanced tree or a k-comet as a spanning tree. Research concerning this and similar questions can be found, for example, in the work of Flandrin et al [5] and Chen et al [4].…”

Section: Introductionmentioning

confidence: 87%

Bounds on the radius and status of graphs

Rissner

Burkard

2014

Networks

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Two classical concepts of centrality in a graph are the median and the center. The connected notions of the status and the radius of a graph seem to be in no relation. In this paper, however, we show a clear connection of both concepts, as they obtain their minimum and maximum values at the same type of tree graphs. Trees with fixed maximum degree and extremum radius and status, resp., are characterized. The bounds on radius and status can be transferred to general connected graphs via spanning trees. A new method of proof allows not only to regain results of Lin et al. on graphs with extremum status, but it allows also to prove analogous results on graphs with extremum radius

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Section: Introductionmentioning

confidence: 87%

Bounds on the radius and status of graphs

Rissner

Burkard

2014

Networks

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“…A spanning tree with at most one branch vertex is called a spider. Gargano, Hammar, Hell, Stacho and Vaccaro [15] (also see Gargano and Hammar [14]) proved that if G is a connected graph on n vertices with δ(G) ≥ (n − 1)/3, then G contains a spanning spider (Later Chen, Ferrara, Hu, Jacobson and Liu [6] proved the stronger result that connected graphs on n ≥ 56 vertices with δ(G) ≥ (n − 2)/3 contain a spanning broom; that is, a spanning spider obtained by joining the center of a star to an endpoint of a path). Motivated by this, Gargano et al [15] conjectured that for all s ≥ 1, if G is a connected graph on n vertices with δ(G) ≥ (n − 1)/(s + 2), then G contains a spanning tree with at most s branch vertices.…”

Section: Introductionmentioning

confidence: 99%

Spanning Trees with Few Branch Vertices

DeBiasio¹,

Lo²

2019

SIAM J. Discrete Math.

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A branch vertex in a tree is a vertex of degree at least three. We prove that, for all s ≥ 1, every connected graph on n vertices with minimum degree at least ( 1 s+3 + o(1))n contains a spanning tree having at most s branch vertices. Asymptotically, this is best possible and solves a problem of Flandrin, Kaiser, Kuzel, Li and Ryjácek, which was originally motivated by an optimization problem in the design of optical networks.

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“…There has been much recent work on conditions implying particular structural properties in spanning trees. See for example , , or the recent survey by Ozeki and Yamashita on spanning trees.…”

Section: Introductionmentioning

confidence: 99%

Degree sum and vertex dominating paths

Faudree

Gould

et al. 2018

Journal of Graph Theory

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A vertex dominating path in a graph is a path P such that every vertex outside P has a neighbor on P. In 1988 H. Broersma [5] stated a result implying that every n‐vertex k‐connected graph G such that σ(k+2)false(Gfalse)≥n−2k−1 contains a vertex dominating path. We provide a short, self‐contained proof of this result and further show that every n‐vertex k‐connected graph such that σ2false(Gfalse)≥2nk+2+ffalse(kfalse) contains a vertex dominating path of length at most (20k)|T|, where T is a minimum dominating set of vertices. An immediate corollary of this result is that every such graph contains a vertex dominating path with length bounded above by a logarithmic function of the order of the graph. To derive this result, we prove that every n‐vertex k‐connected graph with σ2false(Gfalse)≥2nk+2+ffalse(kfalse) contains a path of length at most 20k|T|, through any set of T vertices where false|Tfalse|≤n/900k4.

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Degree Conditions for Spanning Brooms

Cited by 6 publications

References 10 publications

Bounds on the radius and status of graphs

Bounds on the radius and status of graphs

Spanning Trees with Few Branch Vertices

Degree sum and vertex dominating paths

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