## Spanning directed trees with many leaves

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Venue: | SIAM J. Discrete Math |

Citations: | 5 - 4 self |

### BibTeX

@ARTICLE{Alon_spanningdirected,

author = {Noga Alon and Fedor V. Fomin and Gregory Gutin and Michael Krivelevich and Saket Saurabh},

title = {Spanning directed trees with many leaves},

journal = {SIAM J. Discrete Math},

year = {}

}

### OpenURL

### Abstract

Abstract. The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that – every strongly connected n-vertex digraph D with minimum indegree at least 3 has an out-branching with at least (n/4) 1/3 − 1 leaves; – if a strongly connected digraph D does not contain an out-branching with k leaves, then the pathwidth of its underlying graph UG(D) is O(k log k). Moreover, if the digraph is acyclic, the pathwidth is at most 4k. The last result implies that it can be decided in time 2 O(k log2 k) · n O(1) whether a strongly connected digraph on n vertices has an out-branching with at least k leaves. On acyclic digraphs the running time of our algorithm is 2 O(k log k) · n O(1). 1

### Citations

870 | Parameterized Complexity
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- 1999
(Show Context)
Citation Context ...tion of k alone. (For DMLOB such a parameter is the number of leaves in the out-tree.) Problems having such an algorithm are said to be fixed parameter tractable (FPT). The book by Downey and Fellows =-=[17]-=- provides an introduction to the topic of parameterized complexity. For recent developments see the books by Flum and Grohe [23] and by Niedermeier [32]. The parameterized version of DMLOB is defined ... |

359 |
Parameterized Complexity Theory
- Flum, Grohe
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(Show Context)
Citation Context ...d to be fixed parameter tractable (FPT). The book by Downey and Fellows [17] provides an introduction to the topic of parameterized complexity. For recent developments see the books by Flum and Grohe =-=[23]-=- and by Niedermeier [32]. The parameterized version of DMLOB is defined as follows: Given a digraph D and a positive integral parameter k, does D contain an out-branching with at least k leaves? We de... |

284 | Invitation to Fixed-Parameter Algorithms
- Niedermeier
- 2006
(Show Context)
Citation Context ...tractable (FPT). The book by Downey and Fellows [17] provides an introduction to the topic of parameterized complexity. For recent developments see the books by Flum and Grohe [23] and by Niedermeier =-=[32]-=-. The parameterized version of DMLOB is defined as follows: Given a digraph D and a positive integral parameter k, does D contain an out-branching with at least k leaves? We denote the parameterized v... |

234 | Digraphs: Theory, Algorithms and Applications
- Bang-Jensen, Gutin
- 2002
(Show Context)
Citation Context ...llowing simple result gives necessary and sufficient conditions for a digraph to have an out-branching. This assertion allows us to check whether ℓs(D) > 0 in time O(|V (D)| + |A(D)|). Proposition 1 (=-=[7]-=-). A digraph D has an out-branching if and only if D has a unique strong component with no incoming arcs. Let P = u1u2 . . . uq be a directed path in a digraph D. An arc uiuj of D is a forward (backwa... |

194 |
The Probabilistic Method, Second Edition
- Alon, Spencer
- 2000
(Show Context)
Citation Context ... obtained digraph. Observe that G is disconnected and G[V (Q ′ )] and G[V (Q ′′ )] are components of G. Thus, pw(UG(G)) ≤ b, where b = max{pw(UG(G[V (Q ′ )])), pw(UG(G[V (Q ′′ )]))} < 2(t − i + 4.5)k =-=(3)-=- Let (Z1, Z2, . . . , Zr) be a path decomposition of G of width at most b. Then (Z1 ∪ Y, Z2 ∪ Y, . . . , Zr ∪ Y ) is a path decomposition of UG(D[V (Q ′ ) ∪ V (Q ′′ )]) of width at most b + 2k < 2(t −... |

187 |
A linear-time algorithm for finding treedecompositions of small treewidth
- Bodlaender
- 1996
(Show Context)
Citation Context ...trongly connected digraph D. The algorithm is based on a combinatorial bound on the pathwidth of the underlying graph of D. Instead of using results from Section 5, one can use Bodlaender’s algorithm =-=[9]-=- computing (for fixed k) tree decomposition of width k (if such a decomposition exists) in linear time. Combined with our combinatorial bounds this yields a linear time algorithm for k-DMLOB (for a st... |

152 |
Linear time algorithms for NP-hard problems restricted to partial k-trees
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- 1989
(Show Context)
Citation Context ... out-branching of D with at least k leaves or to show that pw(UG(D)) ≤ 4k and provide the corresponding path decomposition. A standard dynamic programming over the path (tree) decomposition (see e.g. =-=[6]-=-) gives us an algorithm of running time 2 O(k log k) · n O(1) . ⊓⊔ The following simple lemma is well known, see, e.g., [15]. Lemma 3. Let T = (V, E) be an undirected tree and let w : V → R + ∪{0} be ... |

85 |
Local Search in Combinatorial Optimization
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(Show Context)
Citation Context ... the pathwidth of its underlying graph UG(D) is O(k log k). Moreover, if the digraph is acyclic, the pathwidth is at most 4k. The last result implies that it can be decided in time 2 O(k log2 k) · n O=-=(1)-=- whether a strongly connected digraph on n vertices has an out-branching with at least k leaves. On acyclic digraphs the running time of our algorithm is 2 O(k log k) · n O(1) . 1 Introduction In this... |

55 |
The vertex separation number of a graph equals its path-width
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- 1992
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Citation Context ...vs(G) = min{vs(G, σ): σ is an ordering of V (G)}. The following assertion is well-known. It follows directly from the results of Kirousis and Papadimitriou [29] on interval width of a graph, see also =-=[28]-=-. Proposition 2 ([28, 29]). For any graph G, vs(G) = pw(G).s3 Locally Optimal Out-Branchings Spanning directed trees with many leaves 5 Our bounds are based on finding locally optimal out-branchings. ... |

37 |
Quickly excluding a forest
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Citation Context ...ves from this minor. Otherwise, there is no K1,k minor in G, and it is possible to prove that the pathwidth of G is O(k). (See, e.g. [8].) Actually, a much more general result due to Bienstock et al. =-=[10]-=-) is that any undirected graph of pathwidth at least k, contains all trees on k vertices as a minor. We prove a result that can be viewed as a generalization of known bounds on the number of leaves in... |

36 |
Spanning trees with many leaves
- Kleitman, West
- 1991
(Show Context)
Citation Context ...nected graph on n vertices with minimum vertex degree δ has a spanning tree with at least n(δ − 2)/(δ + 1) + cδ leaves, where cδ depends on δ. This is indeed the case for all δ ≤ 5. Kleitman and West =-=[27]-=- and Linial and Sturtevant [30] showed that every connected undirected graph G on n vertices with minimum degree at least 3 has a spanning tree with at least n/4 + 2 leaves. Griggs and Wu [25] proved ... |

28 |
Transversal numbers of uniform hypergraphs
- Alon
- 1990
(Show Context)
Citation Context ... . . . , Xs) be a path decomposition of UG(R) of width at most k. Then (X1 ∪ W, X2 ∪ W, . . . , Xs ∪ W ) is a path decomposition of UG(D[V (Q)]) of width less than k + 4k. Thus, pw(UG(D[V (Q)])) < 5k =-=(2)-=- Now assume that we have proved (??) for j = i and show it for j = i − 1. Let Q be a node of layer i − 1. If Q is a leaf of T , we are done by (??). So, we may assume that Q has children Q ′ and Q ′′ ... |

26 | 2-approximation algorithm for finding a spanning tree with the maximum number of leaves
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(Show Context)
Citation Context ...n-degree 3 in which every out-branching has at most O( √ n) leaves. Unlike its undirected counterpart which has attracted a lot of attention in all algorithmic paradigms like approximation algorithms =-=[24, 31, 33]-=-, parameterized algorithms [11, 19, 21], exact exponential time algorithms [20] and also combinatorial studies [16, 25, 27, 30], the Directed Maximum Leaf Out-Branching problem has largely been neglec... |

25 | Approximating maximum leaf spanning trees in almost linear time
- Lu, Ravi
- 1998
(Show Context)
Citation Context ...n-degree 3 in which every out-branching has at most O( √ n) leaves. Unlike its undirected counterpart which has attracted a lot of attention in all algorithmic paradigms like approximation algorithms =-=[24, 31, 33]-=-, parameterized algorithms [11, 19, 21], exact exponential time algorithms [20] and also combinatorial studies [16, 25, 27, 30], the Directed Maximum Leaf Out-Branching problem has largely been neglec... |

22 |
Coordinatized kernels and catalytic reductions: An improved FPT algorithm for max leaf spanning tree and other problems
- Fellows, McCartin, et al.
- 2000
(Show Context)
Citation Context ... has at most O( √ n) leaves. Unlike its undirected counterpart which has attracted a lot of attention in all algorithmic paradigms like approximation algorithms [24, 31, 33], parameterized algorithms =-=[11, 19, 21]-=-, exact exponential time algorithms [20] and also combinatorial studies [16, 25, 27, 30], the Directed Maximum Leaf Out-Branching problem has largely been neglected until recently. The only paper we a... |

18 |
A faster FPT algorithm for finding spanning trees with many leaves
- Bonsma, Brüggemann, et al.
- 2003
(Show Context)
Citation Context ... has at most O( √ n) leaves. Unlike its undirected counterpart which has attracted a lot of attention in all algorithmic paradigms like approximation algorithms [24, 31, 33], parameterized algorithms =-=[11, 19, 21]-=-, exact exponential time algorithms [20] and also combinatorial studies [16, 25, 27, 30], the Directed Maximum Leaf Out-Branching problem has largely been neglected until recently. The only paper we a... |

18 | Solving connected dominating set faster than 2 n
- Fomin, Grandoni, et al.
(Show Context)
Citation Context ...ed counterpart which has attracted a lot of attention in all algorithmic paradigms like approximation algorithms [24, 31, 33], parameterized algorithms [11, 19, 21], exact exponential time algorithms =-=[20]-=- and also combinatorial studies [16, 25, 27, 30], the Directed Maximum Leaf Out-Branching problem has largely been neglected until recently. The only paper we are aware of is the very recent paper [18... |

15 | Connected domination and spanning trees with many leaves
- Caro, West, et al.
(Show Context)
Citation Context ...imum degree δ in ln (δ+1) which no tree has more than (1 − (1 + o(1)) δ+1 )n leaves, where the o(1)-term tends to zero an δ tends to infinity, and this is essentially tight. See also [3], pp. 4-5 and =-=[13]-=- for more information. In this paper we prove an analogue of the Kleitman-West result for directed graphs: every strongly connected digraph D of order n with minimum in-degree at least 3 has an out-br... |

14 |
On linear time minor tests and depth-first search
- Bodlaender
- 1993
(Show Context)
Citation Context ...hen it is possible to construct a spanning tree with at least k leaves from this minor. Otherwise, there is no K1,k minor in G, and it is possible to prove that the pathwidth of G is O(k). (See, e.g. =-=[8]-=-.) Actually, a much more general result due to Bienstock et al. [10]) is that any undirected graph of pathwidth at least k, contains all trees on k vertices as a minor. We prove a result that can be v... |

11 | S.: Parameterized algorithms for directed maximum leaf problems
- Alon, Fomin, et al.
(Show Context)
Citation Context ...anching of D. The vertices of T of out-degree zero are called leaves. The Directed Maximum Leaf Out-Branching (DMLOB) ⋆ Preliminary extended abstracts of this paper have been presented at FSTTCS 2007 =-=[5]-=- and ICALP 2007 [4]s2 N. Alon, F. V. Fomin, G. Gutin, M. Krivelevich, and S. Saurabh problem is to find an out-branching in a given digraph with the maximum number of leaves. The combinatorial study o... |

11 |
Spanning trees with many leaves
- Johnson, Seymour
(Show Context)
Citation Context ... lot of attention in all algorithmic paradigms like approximation algorithms [24, 31, 33], parameterized algorithms [11, 19, 21], exact exponential time algorithms [20] and also combinatorial studies =-=[16, 25, 27, 30]-=-, the Directed Maximum Leaf Out-Branching problem has largely been neglected until recently. The only paper we are aware of is the very recent paper [18] that describes an O( √ opt)-approximation algo... |

10 | S.: Better algorithms and bounds for directed maximum leaf problems
- Alon, Fomin, et al.
(Show Context)
Citation Context ...ertices of T of out-degree zero are called leaves. The Directed Maximum Leaf Out-Branching (DMLOB) ⋆ Preliminary extended abstracts of this paper have been presented at FSTTCS 2007 [5] and ICALP 2007 =-=[4]-=-s2 N. Alon, F. V. Fomin, G. Gutin, M. Krivelevich, and S. Saurabh problem is to find an out-branching in a given digraph with the maximum number of leaves. The combinatorial study of spanning trees wi... |

10 |
On the approximability of some maximum spanning tree problems
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- 1997
(Show Context)
Citation Context ...n-degree 3 in which every out-branching has at most O( √ n) leaves. Unlike its undirected counterpart which has attracted a lot of attention in all algorithmic paradigms like approximation algorithms =-=[24, 31, 33]-=-, parameterized algorithms [11, 19, 21], exact exponential time algorithms [20] and also combinatorial studies [16, 25, 27, 30], the Directed Maximum Leaf Out-Branching problem has largely been neglec... |

9 | Compendium of parameterized problems
- Cesati
- 2005
(Show Context)
Citation Context ...graphs and acyclic digraphs. We remark that the algorithmic results presented here also hold for all digraphs if we consider k-DMLOT rather than k-DMLOB. This answers an open question of Mike Fellows =-=[14, 22, 26]-=-. However, we mainly restrict ourselves to k-DMLOB for clarity and the harder challenges it poses, and we briefly consider k-DMLOT only in the last section. Very recently, using a modification of our ... |

9 |
Private communication
- Fellows
- 2011
(Show Context)
Citation Context ...graphs and acyclic digraphs. We remark that the algorithmic results presented here also hold for all digraphs if we consider k-DMLOT rather than k-DMLOB. This answers an open question of Mike Fellows =-=[14, 22, 26]-=-. However, we mainly restrict ourselves to k-DMLOB for clarity and the harder challenges it poses, and we briefly consider k-DMLOT only in the last section. Very recently, using a modification of our ... |

9 |
Local Search in Combinatorial Optimization. Wiley-Interscience
- Aarts, Lenstra
- 1997
(Show Context)
Citation Context ... the pathwidth of its underlying graph UG(D) is O(k log k). Moreover, if the digraph is acyclic, the pathwidth is at most 4k. The last result implies that it can be decided in time 2 O(k log2 k) · n O=-=(1)-=- whether a strongly connected digraph on n vertices has an out-branching with at least k leaves. On acyclic digraphs the running time of our algorithm is 2 O(k log k) · n O(1) . 1 Introduction In this... |

9 | P.: A new algorithm for finding trees with many leaves
- Kneis, Langer, et al.
- 2008
(Show Context)
Citation Context ...deas and treewidth rather than pathwidth, Bonsma and Dorn [11] designed algorithms of complexity 2 O(k log k) n O(1) for both k-DMLOT and k-DMLOB. Using another approach, Kneis, Langer and Rossmanith =-=[30]-=- obtained an 4 k n O(1) time algorithm for k-DMLOB. It is not difficult to see that this algorithm implies an 4 k n O(1) time algorithm for k-DMLOT. We conclude by pointing out that in a recent paper ... |

8 |
Separator theorems and their applications
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(Show Context)
Citation Context ... A standard dynamic programming over the path (tree) decomposition (see e.g. [6]) gives us an algorithm of running time 2 O(k log k) · n O(1) . ⊓⊔ The following simple lemma is well known, see, e.g., =-=[15]-=-. Lemma 3. Let T = (V, E) be an undirected tree and let w : V → R + ∪{0} be a weight function on its vertices. There exists a vertex v ∈ T such that the weight of every subtree T ′ of T − v is at most... |

8 |
An approximation algorithm for the maximum leaf spanning arborescence problem
- Drescher, Vetta
(Show Context)
Citation Context ...20] and also combinatorial studies [16, 25, 27, 30], the Directed Maximum Leaf Out-Branching problem has largely been neglected until recently. The only paper we are aware of is the very recent paper =-=[18]-=- that describes an O( √ opt)-approximation algorithms for DMLOB. Our second combinatorial result relates the number of leaves in a DMLOB of a directed graph D with the pathwidth of its underlying grap... |

6 |
Spanning trees in graphs of minimum degree four or five. Discrete Mathematics 104
- Griggs, Wu
- 1992
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Citation Context ...nd West [27] and Linial and Sturtevant [30] showed that every connected undirected graph G on n vertices with minimum degree at least 3 has a spanning tree with at least n/4 + 2 leaves. Griggs and Wu =-=[25]-=- proved that the maximum number of leaves in a spanning tree is at least n/2 + 2 when δ = 5 and at least 2n/5 + 8/5 when δ = 4. All these results are tight. The situation is less clear for δ ≥ 6; the ... |

6 | Some Parameterized Problems on Digraphs
- Gutin, Yeo
(Show Context)
Citation Context ...graphs and acyclic digraphs. We remark that the algorithmic results presented here also hold for all digraphs if we consider k-DMLOT rather than k-DMLOB. This answers an open question of Mike Fellows =-=[14, 22, 26]-=-. However, we mainly restrict ourselves to k-DMLOB for clarity and the harder challenges it poses, and we briefly consider k-DMLOT only in the last section. Very recently, using a modification of our ... |

6 | Tight bounds and faster algorithms for Directed Max-Leaf
- Bonsma, Dorn
(Show Context)
Citation Context ...n all digraphs. For acyclic digraphs, the running time can be reduced to 2 O(k log k) · n O(1) . 7 Consequent Research Research initiated by [4] and [5] was continued by Bonsma and Dorn who proved in =-=[11]-=- that every strongly connected digraph of order n with minimum in-degree at least 3 has a out-branching with at least √ n/4 leaves. Thus, the maximum guaranteed number λ(n) of leaves in a strongly con... |

5 | An FPT algorithm for directed spanning k-leaf
- Bonsma, Dorn
- 2007
(Show Context)
Citation Context ...ict ourselves to k-DMLOB for clarity and the harder challenges it poses, and we briefly consider k-DMLOT only in the last section. Very recently, using a modification of our approach, Bonsma and Dorn =-=[12]-=- proved that either an arbitrary digraph D has an out-branching with at most k leaves or the pathwidth of UG(D ′ ) is O(k 3 ), where D ′ is the digraph obtained from D by deleting all arcs not contain... |

1 |
Solving Connected Dominating Set Faster than O(2 n ). Algorithmica 52
- Fomin, Grandoni, et al.
- 2008
(Show Context)
Citation Context ...ntroduction In this paper, we initiate the combinatorial and algorithmic study of a natural generalization of the well studied Maximum Leaf Spanning Tree (MLST) problem on connected undirected graphs =-=[10, 15, 18, 19, 20, 23, 25, 32, 34]-=-. Given a digraph D, a subdigraph T of D is an out-tree if T is an oriented tree with only one vertex s of in-degree zero (called the root). If T is a spanning outtree, i.e. V (T ) = V (D), then T is ... |