scholarly journals A 2-Approximation Algorithm for Finding a Spanning Tree with Maximum Number of Leaves

Algorithmica ◽  
2015 ◽  
Vol 77 (2) ◽  
pp. 374-388 ◽  
Author(s):  
Roberto Solis-Oba ◽  
Paul Bonsma ◽  
Stefanie Lowski
Keyword(s):  
Approximation Algorithm ◽  
Spanning Tree ◽  
Number Of Leaves

On the number of leaves of a euclidean minimal spanning tree

1987 ◽  
Vol 24 (4) ◽  
pp. 809-826 ◽  
Author(s):  
J. Michael Steele ◽  
Lawrence A. Shepp ◽  
William F. Eddy
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Random Variables ◽  
Minimal Spanning Tree ◽  
Absolutely Continuous ◽  
Mean And Variance ◽  
Number Of Leaves ◽  
The Mean ◽  
Vertex Degrees ◽  
Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi, , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n–1Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

On the number of leaves of a euclidean minimal spanning tree

1987 ◽  
Vol 24 (04) ◽  
pp. 809-826 ◽  
Author(s):  
J. Michael Steele ◽  
Lawrence A. Shepp ◽  
William F. Eddy
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Random Variables ◽  
Minimal Spanning Tree ◽  
Absolutely Continuous ◽  
Mean And Variance ◽  
Number Of Leaves ◽  
The Mean ◽  
Vertex Degrees ◽  
Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi , , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n –1 Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

scholarly journals Leafy spanning k-forests

2021 ◽  
Author(s):  
Cristina G. Fernandes ◽  
Carla N. Lintzmayer ◽  
Mário César San Felice
Keyword(s):  
Positive Integer ◽  
Approximation Algorithm ◽  
Best Approximation ◽  
Np Hard ◽  
Connected Graphs ◽  
Spanning Forest ◽  
The Best Approximation ◽  
Number Of Leaves

We denote by Maximum Leaf Spanning k-Forest the problem of, given a positive integer k and a graph G with at most k components, finding a spanning forest in G with at most k components and the maximum number of leaves. A leaf in a forest is defined as a vertex of degree at most one. The case k = 1 for connected graphs is known to be NP-hard, and is well studied in the literature, with the best approximation algorithm proposed more than 20 years ago by Solis-Oba. The best known approximation algorithm for Maximum Leaf Spanning k-Forest with a slightly different leaf definition is a 3-approximation based on an approach by Lu and Ravi for the k = 1 case. We extend the algorithm of Solis-Oba to achieve a 2-approximation for Maximum Leaf Spanning k-Forest.

scholarly journals Np-hardness proof and an approximation algorithm for the minimum vertex ranking spanning tree problem

2006 ◽  
Vol 154 (16) ◽  
pp. 2402-2410 ◽  
Author(s):  
Keizo Miyata ◽  
Shigeru Masuyama ◽  
Shin-ichi Nakayama ◽  
Liang Zhao
Keyword(s):  
Approximation Algorithm ◽  
Spanning Tree ◽  
Vertex Ranking ◽  
Np Hardness

scholarly journals Spanning Trees with Many Leaves and Average Distance

10.37236/757 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ermelinda DeLaViña ◽  
Bill Waller
Keyword(s):  
Lower Bounds ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Average Distance ◽  
Independence Number ◽  
Local Independence ◽  
Open Problems ◽  
Maximum Order ◽  
Bipartite Subgraph ◽  
Number Of Leaves

In this paper we prove several new lower bounds on the maximum number of leaves of a spanning tree of a graph related to its order, independence number, local independence number, and the maximum order of a bipartite subgraph. These new lower bounds were conjectured by the program Graffiti.pc, a variant of the program Graffiti. We use two of these results to give two partial resolutions of conjecture 747 of Graffiti (circa 1992), which states that the average distance of a graph is not more than half the maximum order of an induced bipartite subgraph. If correct, this conjecture would generalize conjecture number 2 of Graffiti, which states that the average distance is not more than the independence number. Conjecture number 2 was first proved by F. Chung. In particular, we show that the average distance is less than half the maximum order of a bipartite subgraph, plus one-half; we also show that if the local independence number is at least five, then the average distance is less than half the maximum order of a bipartite subgraph. In conclusion, we give some open problems related to average distance or the maximum number of leaves of a spanning tree.

scholarly journals An Approximation Algorithm for Maximum Internal Spanning Tree

2017 ◽  
pp. 385-396
Author(s):  
Zhi-Zhong Chen ◽  
Youta Harada ◽  
Fei Guo ◽  
Lusheng Wang
Keyword(s):  
Approximation Algorithm ◽  
Spanning Tree

A 43-approximation algorithm for the Maximum Internal Spanning Tree Problem

2021 ◽  
Vol 118 ◽  
pp. 131-140
Author(s):  
Xingfu Li ◽  
Daming Zhu ◽  
Lusheng Wang
Keyword(s):  
Approximation Algorithm ◽  
Spanning Tree

scholarly journals Listing all spanning trees in Halin graphs — sequential and Parallel view

2018 ◽  
Vol 10 (01) ◽  
pp. 1850005
Author(s):  
K. Krishna Mohan Reddy ◽  
P. Renjith ◽  
N. Sadagopan
Keyword(s):  
Parallel Algorithm ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Combinatorial Problem ◽  
Halin Graph ◽  
Labeled Graph ◽  
Number Of Leaves ◽  
Acyclic Subgraph ◽  
Number Of Spanning Trees ◽  
Halin Graphs

For a connected labeled graph [Formula: see text], a spanning tree [Formula: see text] is a connected and acyclic subgraph that spans all vertices of [Formula: see text]. In this paper, we consider a classical combinatorial problem which is to list all spanning trees of [Formula: see text]. A Halin graph is a graph obtained from a tree with no degree two vertices and by joining all leaves with a cycle. We present a sequential and parallel algorithm to enumerate all spanning trees in Halin graphs. Our approach enumerates without repetitions and we make use of [Formula: see text] processors for parallel algorithmics, where [Formula: see text] and [Formula: see text] are the depth, the number of leaves, respectively, of the Halin graph. We also prove that the number of spanning trees in Halin graphs is [Formula: see text].

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