A degree bound on decomposable trees

Barth, Dominique; Fournier, Hervé

doi:10.1016/j.disc.2006.01.006

Cited by 36 publications

(39 citation statements)

References 8 publications

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“…In [176] Horňák and Woźniak conjectured that if T is a tree with maximum degree ∆(T ) at least five, then T is not avd. This conjecture was proved by Barth and Fournier [21].…”

Section: Arbitrarily Vertex Decomposable Graphsmentioning

confidence: 76%

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Cycles in graphs and related problems

Marczyk

2008

Dissertationes Math.

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“…In [176] Horňák and Woźniak conjectured that if T is a tree with maximum degree ∆(T ) at least five, then T is not avd. This conjecture was proved by Barth and Fournier [21].…”

Section: Arbitrarily Vertex Decomposable Graphsmentioning

confidence: 76%

“…In [20] and [21] Barth et al and Barth and Fournier studied a family of trees each of them being homeomorphic to K 1,3 or K 1,4 (they call them tripods or 4-pods) and showed that determining if such a tree is avd can be done using a polynomial algorithm.…”

Section: Arbitrarily Vertex Decomposable Graphsmentioning

confidence: 99%

Cycles in graphs and related problems

Marczyk

2008

Dissertationes Math.

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“…The investigation of AP trees gained lots of attention in this context, since a connected graph is AP if one of its spanning trees is AP. It turned out, however, that the structure of AP trees is not obvious in general (see for instance [3], [4], [5] or [14]). …”

Section: Introductionmentioning

confidence: 99%

Partitioning powers of traceable or hamiltonian graphs

Baudon

Bensmail

Przybyło

et al. 2014

Theoretical Computer Science

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A graph G = (V, E) is arbitrarily partitionable (AP) if for any sequence τ = (n 1 , . . . , n p ) of positive integers adding up to the order of G, there is a sequence of mutually disjoints subsets of V whose sizes are given by τ and which induce connected graphs. If, additionally, for given k, it is possible to prescribe l = min{k, p} vertices belonging to the first l subsets of τ , G is said to be AP+k.The paper contains the proofs that the k th power of every traceable graph of order at least k is AP+(k − 1) and that the k th power of every hamiltonian graph of order at least 2k is AP+(2k −1), and these results are tight.

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“…There are various results characterizing different families of avd trees [1,2,5,7,8]. The complete characterization of on-line avd trees was recently found by Horňák et al [9].…”

Section: Introductionmentioning

confidence: 99%

“…The most important result on avd trees due to Barth and Fournier [2] (conjectured by Horňák and Woźniak [8]) states that if T is a tree with maximum degree (T ) at least five, then T is not avd.…”

Section: Introductionmentioning

confidence: 99%

Dense Arbitrarily Vertex Decomposable Graphs

Horňák

Marczyk

Schiermeyer

et al. 2011

Graphs and Combinatorics

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A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n 1 , . . . , n k ) of positive integers such that n 1 + · · · + n k = n there exists a partition (V 1 , . . . , V k ) of the vertex set of G such that for each i ∈ {1, . . . , k}, V i induces a connected subgraph of G on n i vertices. The main result of the paper reads as follows. Suppose that G is a connected graph on n ≥ 20 vertices that admits a perfect matching or a matching omitting exactly one vertex. If the degree sum of any pair of nonadjacent vertices is at least n − 5, then G is arbitrarily vertex decomposable. We also describe 2-connected arbitrarily vertex decomposable graphs that satisfy a similar degree sum condition.

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A degree bound on decomposable trees

Cited by 36 publications

References 8 publications

Cycles in graphs and related problems

Cycles in graphs and related problems

Partitioning powers of traceable or hamiltonian graphs

Dense Arbitrarily Vertex Decomposable Graphs

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