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### The Connectivities of Leaf Graphs of Sets of Points in the Plane

Atsushi KANEKO, Kiyoshi YOSHIMOTO
2001 Tokyo Journal of Mathematics
Let U be a set of n points in the plane. If no three points are collinear, then we s a y  ...  Broersma and Li Xueliang, The connectivity of the leaf-exchange ane spanning tree graph of a graph, ARS Comb. 43 (1996) , 225-231  E. Campo and V.  ...  In this paper, we study a spanning subgraph of the tree graph on U, called a leaf graph.  ...

### Enumeration of Spanning Trees Using Edge Exchange with Minimal Partitioning [article]

Nasr Mohamed
2014 arXiv   pre-print
While MP algorithm uses a computational tree graph to traverse all possible spanning trees by the edge exchange technique, it has two unique properties compared to previous algorithms.  ...  Practically, and as a result of this property, the interface between the two partitions of the spanning tree during edge exchange has to be maintained from one side only.  ...  (a) shows a graph The parent of a type 2 spanning tree is the spanning tree formed by exchanging the leaf pilot pair of the tree with the leaf pair that has the same node component and with edge that has  ...

### Page 2775 of Mathematical Reviews Vol. , Issue 97E [page]

1997 Mathematical Reviews
A variation on the leaf-exchange spanning tree graph appeared in recent work on basis graphs of branching greedoids.  ...  We characterize the graphs which have a connected leaf-exchange spanning tree graph and give a lower bound on the connectivity of 7)(G) for a 3-connected graph G.” 97e:05129 05C40 05C38 Gao, Jingzhen (  ...

### Spanning Trees: A Survey

Kenta Ozeki, Tomoki Yamashita
2010 Graphs and Combinatorics
In this paper, we give a survey of spanning trees.  ...  We mainly deal with spanning trees having some particular properties concerning a hamiltonian properties, for example, spanning trees with bounded degree, with bounded number of leaves, or with bounded  ...  Acknowledgements The authors would like to thank the referees for their helpful comments and corrections.  ...

### On the Structure of the Graph of Unique Symmetric Base Exchanges of Bispanning Graphs [article]

Timo Bingmann
2016 arXiv   pre-print
The operation of symmetrically swapping two edges between the trees, such that the result is a different pair of disjoint spanning trees, is called an edge exchange or a symmetric base exchange.  ...  The graph of symmetric base exchanges of a bispanning graph contains a vertex for every valid pair of disjoint spanning trees, and edges between them to represent all possible edge exchanges.  ...  Lemma 4 . 4 13 (leafUEs: unique exchanges due to leaf edges) If (S, T ) ∈ V τ (G) is a pair of disjoint spanning trees of a bispanning graph G = (V, E, δ), and v ∈ V is a leaf in the tree-graph G[S] incident  ...

### Fixed Parameter Evolutionary Algorithms and Maximum Leaf Spanning Trees: A Matter of Mutation [chapter]

Stefan Kratsch, Per Kristian Lehre, Frank Neumann, Pietro Simone Oliveto
2010 Parallel Problem Solving from Nature, PPSN XI
We investigate the NP-hard problem of computing a spanning tree that has a maximal number of leaves by evolutionary algorithms in the context of fixed parameter tractability (FPT) where the maximum number  ...  Investigating two common mutation operators, we show that an operator related to spanning tree problems leads to an FPT running time in contrast to a general mutation operator that does not have this property  ...  Given an undirected connected graph G = (V, E), the goal is to find a spanning tree T * of G such that the number of leaves is maximal.  ...

### The connectivity of the basis graph of a branching greedoid

H. J. Broersma, Li Xueliang
1992 Journal of Graph Theory
A result of Korte and L O V~S Z states that the basis graph of every 2connected greedoid is connected.  ...  We prove that the basis graph of every 3-connected branching greedoid is (6 -1)-connected, where 6 is the minimum in-degree (disregarding the root) of the underlying rooted directed (mu1ti)graph.  ...  This also implies that every spanning tree of a 2-connected rooted graph can be changed into a given second one by successively exchanging a single leaf in every step.  ...

### A 2k-Vertex Kernel for Maximum Internal Spanning Tree [article]

Wenjun Li, Jianxin Wang, Jianer Chen, Yixin Cao
2014 arXiv   pre-print
We consider the parameterized version of the maximum internal spanning tree problem, which, given an n-vertex graph and a parameter k, asks for a spanning tree with at least k internal vertices.  ...  Our algorithm applies first a greedy procedure consisting of a sequence of local exchange operations, which ends with a local-optimal spanning tree, and then uses this special tree to find a reducible  ...  Introduction A spanning tree of a connected graph G is a subgraph that includes all the vertices of G and is a tree.  ...

### An approximation algorithm for the maximum leaf spanning arborescence problem

2010 ACM Transactions on Algorithms
We present an O( √ opt)-approximation algorithm for the maximum leaf spanning arborescence problem, where opt is the number of leaves in an optimal spanning arborescence.  ...  The result is based upon an O(1)-approximation algorithm for a special class of directed graphs called willows.  ...  A natural question is then to find a spanning tree T in a undirected graph G containing the maximum number of leaves.  ...

### The Connectivities of Leaf Graphs of 2-Connected Graphs

Atsushi Kaneko, Kiyoshi Yoshimoto
1999 Journal of combinatorial theory. Series B (Print)
Given a connected graph G, denote by V the family of all the spanning trees of G.  ...  Define an adjacency relation in V as follows: the spanning trees t and t\$ are said to be adjacent if for some vertex u # V, t&u is connected and coincides with t\$&u.  ...  ACKNOWLEDGMENTS The authors thank Professor S. Matsumoto and Professor Y. Egawa for his helpful advice.  ...

### Spanning tree with many leaves in quadratic graph

Akanksa Rastogi, Vinay Singhal
2013 International Journal of Research Studies in Computing
For a connected graph G let L(G) denote the maximum number of leaves in any spanning tree of G.  ...  Another large class of properties deals with the structure of the leaves of the spanning tree.  ...  As we know tree is a connected graph, in the last steps of this algorithm we connect T and the isolated vertices (which does not satisfy inequality) and return the connected spanning tree TA.  ...

### Distributed algorithms for finding and maintaining a k-tree core in a dynamic network

Saurabh Srivastava, R.K. Ghosh
2003 Information Processing Letters
In this paper first we propose a distributed algorithm for constructing a rooted spanning tree of a dynamic graph such that root of the tree is located near the center of the graph.  ...  In the next stage these trees are connected to form a single rooted tree.  ...  For a graph of this kind finding a rooted spanning tree itself can be challenging.  ...

### An artificial bee colony algorithm for the leaf-constrained minimum spanning tree problem

Alok Singh
2009 Applied Soft Computing
Introduction Given an undirected, connected, weighted graph with n nodes and a positive integer 'ð2 ' < n À 1Þ, the leaf-constrained minimum spanning tree (LCMST) problem seeks on this graph a spanning  ...  Julstrom  observed that for ' ¼ 0:6n, ML finds lower-weight Given an undirected, connected, weighted graph, the leaf-constrained minimum spanning tree (LCMST) problem seeks on this graph a spanning  ...

### The Chartrand-Schuster conjecture: Graphs with unique distance trees are regular

Sukhamay Kundu
1977 Journal of combinatorial theory. Series B (Print)
A distance tree T is a spanning tree of G which further satisfies the condition that for some vertex E, d&, u) = dr(u, u) for aI U, where d&v, u) denotes the distance of u from v in the graph G.  ...  Let G be a finite connected graph with no cut vertex.  ...  Let T1 and T, be two distance trees at a vertex x of a connected graph G. Then one can pass from T1 to T, in a\$nite sequence of "exchanges." Proof. Assume T, # T, .  ...

### Reconfiguration of Spanning Trees with Many or Few Leaves

Nicolas Bousquet, Takehiro Ito, Yusuke Kobayashi, Haruka Mizuta, Paul Ouvrard, Akira Suzuki, Kunihiro Wasa, Peter Sanders, Fabrizio Grandoni, Grzegorz Herman
2020 European Symposium on Algorithms
Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed in [Takehiro Ito et al., 2011].  ...  Let G be a graph and T₁,T₂ be two spanning trees of G. We say that T₁ can be transformed into T₂ via an edge flip if there exist two edges e ∈ T₁ and f in T₂ such that T₂ = (T₁⧵e) ∪ f.  ...  A spanning tree T of G is frozen if all the spanning trees in its connected component of the reconfiguration graph have the same internal nodes. Claim 22 (*). Let T be a spanning tree of G.  ...
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