The number of spanning trees of a graph with given matching number

The number of spanning trees of a graph G is the total number of distinct spanning subgraphs of G that are trees. In this paper, we present sharp upper bounds for the number of spanning trees of a graph with given matching number.     

The number of spanning trees of the regular networks

The classic theorem on graphs and matrices is the Matrix-Tree Theorem, which gives the number of spanning trees t(G) of any graph G as the value of a certain determinant. However, in this paper, we will derive a simple formula for the number of spanning trees of the regular networks.     

An Algorithm for Enumerating All Spanning Trees of a Directed Graph

We present an O(NV + V ³ ) time algorithm for enumerating all spanning trees of a directed graph. This improves the previous best known bound of O(NE + V+E) [1] when V ² =o(N) , which will be true for most graphs. Here, N refers to the number of spanning trees of a graph having V vertices and E edges. The algorithm is based on the technique of obtaining one spanning tree from another by a series of edge swaps. This result complements the result in the companion paper [3] which enumerates all spanning trees in an undirected graph in O(N+V+E) time. Received September 11, 1997; revised March 6, 1998.     

Listing all the minimum spanning trees in an undirected graph

Efficient polynomial time algorithms are well known for the minimum spanning tree problem. However, given an undirected graph with integer edge weights, minimum spanning trees may not be unique. In this article, we present an algorithm that lists all the minimum spanning trees included in the graph. The computational complexity of the algorithm is O(N(mn+n ² log n)) in time and O(m) in space, where n, m and N stand for the number of nodes, edges and minimum spanning trees, respectively. Next, we explore some properties of cut-sets, and based on these we construct an improved algorithm, which runs in O(N m log n) time and O(m) space. These algorithms are implemented in C language, and some numerical experiments are conducted for planar as well as complete graphs with random edge weights.     

A Flexible Algorithm for Generating All the Spanning Trees in Undirected Graphs

In this paper we propose an algorithm for generating all the spanning trees in undirected graphs. The algorithm requires O (n+m+ τ n) time where the given graph has n vertices, m edges, and τ spanning trees. For outputting all the spanning trees explicitly, this time complexity is optimal. Our algorithm follows a special rooted tree structure on the skeleton graph of the spanning tree polytope. The rule by which the rooted tree structure is traversed is irrelevant to the time complexity. In this sense, our algorithm is flexible. If we employ the depth-first search rule, we can save the memory requirement to O (n+m). A breadth-first implementation requires as much as O (m+ τ n) space, but when a parallel computer is available, this might have an advantage. When a given graph is weighted, the best-first search rule provides a ranking algorithm for the minimum spanning tree problem. The ranking algorithm requires O (n+ m + τ n) time and O (m+ τ n) space when we have a minimum spanning tree. Received January 21, 1995; revised February 19, 1996.     

<Emphasis Type="Italic">k</Emphasis>-Outerplanar Graphs,Planar Duality,and Low Stretch Spanning Trees

Low distortion probabilistic embedding of graphs into approximating trees is an extensively studied topic. Of particular interest is the case where the approximating trees are required to be (subgraph) spanning trees of the given graph (or multigraph), in which case, the focus is usually on the equivalent problem of finding a (single) tree with low average stretch. Among the classes of graphs that received special attention in this context are k-outerplanar graphs (for a fixed k): Chekuri, Gupta, Newman, Rabinovich, and Sinclair show that every k-outerplanar graph can be probabilistically embedded into approximating trees with constant distortion regardless of the size of the graph. The approximating trees in the technique of Chekuri et al. are not necessarily spanning trees, though.     

Improved heuristics for the bounded-diameter minimum spanning tree problem

Given an undirected, connected, weighted graph and a positive integer k, the bounded-diameter minimum spanning tree (BDMST) problem seeks a spanning tree of the graph with smallest weight, among all spanning trees of the graph, which contain no path with more than k edges. In general, this problem is NP-Hard for 4 ≤ k < n − 1, where n is the number of vertices in the graph. This work is an improvement over two existing greedy heuristics, called randomized greedy heuristic (RGH) and centre-based tree construction heuristic (CBTC), and a permutation-coded evolutionary algorithm for the BDMST problem. We have proposed two improvements in RGH/CBTC. The first improvement iteratively tries to modify the bounded-diameter spanning tree obtained by RGH/CBTC so as to reduce its cost, whereas the second improves the speed. We have modified the crossover and mutation operators and the decoder used in permutation-coded evolutionary algorithm so as to improve its performance. Computational results show the effectiveness of our approaches. Our approaches obtained better quality solutions in a much shorter time on all test problem instances considered.     

Stability analysis of constrained MPC with CLF applied to discrete-time nonlinear system

The problem of counting the number of spanning trees is an old topic in graph theory with important applications to reliable network design. Usually, it is desirable to put forward a formula of the number of spanning trees for various graphs, which is not only interesting in its own right but also in practice. Since some large graphs can be composed of some existing smaller graphs by using the product of graphs, the number of spanning trees of such large graph is also closely related to that of the corresponding smaller ones. In this article, we establish a formula for the number of spanning trees in the lexicographic product of two graphs, in which one graph is an arbitrary graph G and the other is a complete multipartite graph. The results extend some of the previous work, which is closely related to the number of vertices and Lapalacian eigenvalues of smaller graphs only.     

Efficient parallel algorithms for breadth first spanning forests of general graphs

Ghosh and Bhattacharjee propose [2] (Intern. J. Computer Math., 1984, Vol. 15, pp. 255-268) an algorithm of determining breadth first spanning trees for graphs, which requires that the input graphs contain some vertices, from which every other vertex in the input graph can be reached. These vertices are called starting vertices. The complexity of the GB algorithm is O(log² n) using O{n ³) processors. In this paper an algorithm, named BREADTH, also computing breadth first spanning trees, is proposed. The complexity is O(log² n) using O{n ³/logn) processors. Then an efficient parallel algorithm, named- BREADTHFOREST, is proposed, which generalizes algorithm BREADTH. The output of applying BREADTHFOREST to a general graph, which may not contain any starting vertices, is a breadth first spanning forest of the input graph. The complexity of BREADTHFOREST is the same as BREADTH.     

On maximally distant spanning trees of a graph

A set ofk spanning trees of a graphG is calledmaximally distant if their union contains the maximum number of edges ofG. We present a necessary and sufficient condition for a set of spanning trees to be maximally distant. We also give an efficient algorithm which actually findsk maximally distant spanning trees in a given graph.     

Independent spanning trees on twisted cubes

Multiple independent spanning trees have applications to fault tolerance and data broadcasting in distributed networks. There are two versions of the n independent spanning trees conjecture. The vertex (edge) conjecture is that any n-connected (n-edge-connected) graph has n vertex-independent spanning trees (edge-independent spanning trees) rooted at an arbitrary vertex. Note that the vertex conjecture implies the edge conjecture. The vertex and edge conjectures have been confirmed only for n-connected graphs with n≤4, and they are still open for arbitrary n-connected graph when n≥5. In this paper, we confirm the vertex conjecture (and hence also the edge conjecture) for the n-dimensional twisted cube TQ_n by providing an O(NlogN) algorithm to construct n vertex-independent spanning trees rooted at any vertex, where N denotes the number of vertices in TQ_n. Moreover, all independent spanning trees rooted at an arbitrary vertex constructed by our construction method are isomorphic and the height of each tree is n+1 for any integer n≥2.     

Independent spanning trees vs. edge-disjoint spanning trees in locally twisted cubes

Fault-tolerant broadcasting and secure message distribution are important issues for numerous applications in networks. It is a common idea to design multiple spanning trees with a specific property in the underlying graph of a network to serve as a broadcasting scheme or a distribution protocol for receiving high levels of fault-tolerance and of security. A set of spanning trees in a graph is said to be edge-disjoint if these trees are rooted at the same node without sharing any common edge. Hsieh and Tu [S.-Y. Hsieh, C.-J. Tu, Constructing edge-disjoint spanning trees in locally twisted cubes, Theoretical Computer Science 410 (2009) 926-932] recently presented an algorithm for constructing n edge-disjoint spanning trees in an n-dimensional locally twisted cube. In this paper, we prove that indeed all spanning trees constructed by their algorithm are independent, i.e., any two spanning trees are rooted at the same node, say r, and for any other node v≠r, the two different paths from v to r, one path in each tree, are internally node-disjoint.     

The number of spanning trees in a new lexicographic product of graphs

The problem of counting the number of spanning trees is an old topic in graph theory with important applications to reliable network design. Usually, it is desirable to put forward a formula of the number of spanning trees for various graphs, which is not only interesting in its own right but also in practice. Since some large graphs can be composed of some existing smaller graphs by using the product of graphs, the number of spanning trees of such large graph is also closely related to that of the corresponding smaller ones. In this article, we establish a formula for the number of spanning trees in the lexicographic product of two graphs, in which one graph is an arbitrary graph G and the other is a complete multipartite graph. The results extend some of the previous work, which is closely related to the number of vertices and Lapalacian eigenvalues of smaller graphs only.     

Spanning Trees and the Complexity of Flood-Filling Games

We consider problems related to the combinatorial game (Free-) Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of flooding operations. We show that the minimum number of moves required to flood any given graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T. This result is then applied to give two polynomial-time algorithms for flood-filling problems. Firstly, we can compute in polynomial time the minimum number of moves required to flood a graph with only a polynomial number of connected subgraphs. Secondly, given any coloured connected graph and a subset of the vertices of bounded size, the number of moves required to connect this subset can be computed in polynomial time.     

Cyclic bundle Hamiltonicity

Cyclic bundle Hamiltonicity cbH(G) of a graph G is the minimal n for which there is an automorphism α of G such that the graph bundle C _n □^α G is Hamiltonian. We define an invariant I that is related to the maximal vertex degree of spanning trees suitably involving the symmetries of G and prove cbH(G)≤I≤cbH(G)+1 for any non-trivial connected graph G.     

Maximizing the number of spanning trees of networks based on cycle basis representation

In this paper, we consider the problem of finding the graph topology of p vertices and q edges such that the resulting graph contains the maximum number of spanning trees. For any values of p and q, this problem still remains open. Only when q≧ p(p? l)/2 ? [p/2] it has been solved by Shier [4] and, recently, when q = p + 1 it has been solved by Wang and Wu [5]. In this paper, we shall formulate this problem into a nonlinear integer program by using the properties of the cycle bases of graphs. And then we shall show that this nonlinear integer program can be easily solved when q = p + 1 by applying our cycle basis approach. Consequently, we shall solve the nonlinear integer program when q = p + 2. Finally, concluding remarks will be given.     

Maximizing the number of spanning trees of networks based on cycle basis representation

In this paper, we consider the problem of finding the graph topology of p vertices and q edges such that the resulting graph contains the maximum number of spanning trees. For any values of p and q, this problem still remains open. Only when q≧ p(p? l)/2 ? [p/2] it has been solved by Shier [4] and, recently, when q = p+1 it has been solved by Wang and Wu [5]. In this paper, we shall formulate this problem into a nonlinear integer program by using the properties of the cycle bases of graphs. And then we shall show that this nonlinear integer program can be easily solved when q = p+ 1 by applying our cycle basis approach. Consequently, we shall solve the nonlinear integer program when q =p + 2. Finally, concluding remarks will be given.     

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

A k-core C_k of a tree T is subtree with exactly k leaves for k?n_l, where n_l the number of leaves in T, and minimizes the sum of the distances of all nodes from C_k. 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. Then we provide a distributed algorithm for finding k-core of that spanning tree. The spanning tree is constructed in two stages. In the first stage, a forest of trees is generated. In the next stage these trees are connected to form a single rooted tree. An interesting aspect of the first stage of proposed spanning algorithm is that it implicitly constructs the (convex) hull of those nodes which are not already included in the spanning forest. The process is repeated till all non root nodes of the graph have chosen a unique parent. We implemented the algorithms for finding spanning tree and its k-core. A core can be quite useful for routing messages in a dynamic network consisting of a set of mobile devices.     

A hierarchy of the class of apex NLC graph languages by bounds on the number of nonterminal nodes in productions

 In this paper, we consider derivation trees in apex NLC graph languages. We show that for every graph H in arbitrary apex NLC graph language, a decomposition tree of H can be constructed from a derivation tree of H. We also show that there exists a hierarchy in the class of apex NLC graph languages when we restrict the number of nonterminals in the right-hand sides of productions in apex NLC graph grammars. Received: 5 July 1994/12 February 1996     

Optimal Independent Spanning Trees on Odd Graphs

The use of multiple independent spanning trees (ISTs) for data broadcasting in networks provides a number of advantages, including the increase of fault-tolerance and bandwidth. The designs of multiple ISTs on several classes of networks have been widely investigated. In this paper we show a construction algorithm of ISTs on odd graphs, and we analyze that all the lengths of the paths in the ISTs are less than or equal to the length of the shortest path+4, which is optimal. We also prove that the heights of the ISTs we constructed are d+1, which again is optimal, since the fault diameter of an odd graph is d+1.