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.     

字典乘积网络的支撑树计数

支撑树个数是边失效下网络可靠性分析与设计的一个重要性能参考指标,本文利用字典乘积的方法来构建网络,通过这种方法我们很容易由若干特定规模较小网络来构建规模较大的网络,并得到它的一个紧的支撑树计数解析公式,这样的计数公式仅仅依赖于小网络的性能参数,如:结点的度数、小网络的阶数、小网络的支撑树数目.     

Variations of the maximum leaf spanning tree problem for bipartite graphs

The maximum leaf spanning tree problem is known to be NP-complete. In [M.S. Rahman, M. Kaykobad, Complexities of some interesting problems on spanning trees, Inform. Process. Lett. 94 (2005) 93-97], a variation on this problem was posed. This variation restricts the problem to bipartite graphs and asks, for a fixed integer K, whether or not the graph contains a spanning tree with at least K leaves in one of the partite sets. We show not only that this problem is NP-complete, but that it remains NP-complete for planar bipartite graphs of maximum degree 4. We also consider a generalization of a related decision problem, which is known to be polynomial-time solvable. We show the problem is still polynomial-time solvable when generalized to weighted graphs.     

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.     

一类多重字典乘积网络的支撑树计数

网络的支撑树个数是衡量一个网络可靠性程度的重要参考指标. 利用字典乘积方法设计的网络, 在应用数学与网络优化设计与分析领域变的重要起来．本文利用组合方法给出了一类新网络的支撑树计数公式,它仅仅依赖小网络的结构拓扑参数：阶数,拉谱拉斯特征值等．     

Locally connected spanning trees in cographs, complements of bipartite graphs and doubly chordal graphs

A spanning tree T of a graph G=(V,E) is called a locally connected spanning tree if the set of all neighbors of v in T induces a connected subgraph of G for all v∈V. The problem of recognizing whether a graph admits a locally connected spanning tree is known to be NP-complete even when the input graphs are restricted to chordal graphs. In this paper, we propose linear time algorithms for finding locally connected spanning trees in cographs, complements of bipartite graphs and doubly chordal graphs, respectively.     

Fast processing of graph queries on a large database of small and medium-sized data graphs

We propose a new way of indexing a large database of small and medium-sized graphs and processing exact subgraph matching (or subgraph isomorphism) and approximate (full) graph matching queries. Rather than decomposing a graph into smaller units (e.g., paths, trees, graphs) for indexing purposes, we represent each graph in the database by its graph signature, which is essentially a multiset. We construct a disk-based index on all the signatures via bulk loading. During query processing, a query graph is also mapped into its signature, and this signature is searched using the index by performing multiset operations. To improve the precision of exact subgraph matching, we develop a new scheme using the concept of line graphs. Through extensive evaluation on real and synthetic graph datasets, we demonstrate that our approach provides a scalable and efficient disk-based solution for a large database of small and medium-sized graphs.     

Theoretical underpinnings for maximal clique enumeration on perturbed graphs

The problem of enumerating the maximal cliques of a graph is a computationally expensive problem with applications in a number of different domains. Sometimes the benefit of knowing the maximal clique enumeration (MCE) of a single graph is worth investing the initial computation time. However, when graphs are abstractions of noisy or uncertain data, the MCE of several closely related graphs may need to be found, and the computational cost of doing so becomes prohibitively expensive.Here, we present a method by which the cost of enumerating the set of maximal cliques for related graphs can be reduced. By using the MCE for some baseline graph, the MCE for a modified, or perturbed, graph may be obtained by enumerating only the maximal cliques that are created or destroyed by the perturbation. When the baseline and perturbed graphs are relatively similar, the difference set between the two MCEs can be overshadowed by the maximal cliques common to both. Thus, by enumerating only the difference set between the baseline and perturbed graphs’ MCEs, the computational cost of enumerating the maximal cliques of the perturbed graph can be reduced.We present necessary and sufficient conditions for enumerating difference sets when the perturbed graph is formed by several different types of perturbations. We also present results of an algorithm based on these conditions that demonstrate a speedup over traditional calculations of the MCE of perturbed, real biological networks.     

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.     

Parallel construction of optimal independent spanning trees on Cartesian product of complete graphs

A set of k spanning trees rooted at the same vertex r in a graph G is said to be independent if for each vertex x other than r, the k paths from r to x, one path in each spanning tree, are internally disjoint. Using independent spanning trees (ISTs) one can design fault-tolerant broadcasting schemes and increase message security in a network. Thus, the problem of ISTs on graphs has been received much attention. Recently, Yang et al. proposed a parallel algorithm for generating optimal ISTs on the hypercube. In this paper, we propose a similar algorithm for generating optimal ISTs on Cartesian product of complete graphs. The algorithm can be easily implemented in parallel or distributed systems. Moreover, the proof of its correctness is simpler than that of Yang et al.     

Parameterized Complexity of the Spanning Tree Congestion Problem

We study the problem of determining the spanning tree congestion of a?graph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k. We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k??8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe that for k??3 the problem becomes polynomially time solvable.     

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.     

<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.     

Maximizing the number of spanning trees on the (p,p+2) graphs

This paper considers the problem of maximizing the number of spanning trees. A newly established result is the formula and the graph topology for the maximum number of spanning trees among the class of (p, p+2) graph.     

Embedding k(n−k) edge-disjoint spanning trees in arrangement graphs

The arrangement graphs are a class of generalized star graphs. In this paper we construct a graph that consists of the maximum number of directed edge-disjoint spanning trees in an arrangement graph. The paths that connect the common root node to any given node through different spanning trees are node-disjoint, and the lengths of these paths differ from the shortest possible lengths by a small additive constant. This graph can be used to derive fault-tolerant algorithms for broadcasting and scattering problems without prior knowledge of the faulty elements of the network.     

On the number of spanning trees of circulant graphs

In this paper, we derive a simple formula for the number of spanning trees of the circulant graphs. Some special cases of the circulant graphs are also taken into account.     

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.     

Expected parallel time and sequential space complexity of graph and digraph problems

This paper determines upper bounds on the expected time complexity for a variety of parallel algorithms for undirected and directed random graph problems. For connectivity, biconnectivity, transitive closure, minimum spanning trees, and all pairs minimum cost paths, we prove the expected time to beO(log logn) for the CRCW PRAM (this parallel RAM machine allows resolution of write conflicts) andO(logn · log logn) for the CREW PRAM (which allows simultaneous reads but not simultaneous writes). We also show that the problem of graph isomorphism has expected parallel timeO(log logn) for the CRCW PRAM andO(logn) for the CREW PRAM. Most of these results follow because of upper bounds on the mean depth of a graph, derived in this paper, for more general graphs than was known before. For undirected connectivity especially, we present a new probabilistic algorithm which runs on a randomized input and has an expected running time ofO(log logn) on the CRCW PRAM, withO(n) expected number of processors only. Our results also improve known upper bounds on the expected space required for sequential graph algorithms. For example, we show that the problems of finding connected components, transitive closure, minimum spanning trees, and minimum cost paths have expected sequential spaceO(logn · log logn) on a deterministic Turing Machine. We use a simulation of the CRCW PRAM to get these expected sequential space bounds.     

Types of arcs in a fuzzy graph

The concept of connectivity plays an important role in both theory and applications of fuzzy graphs. Depending on the strength of an arc, this paper classifies arcs of a fuzzy graph into three types namely α-strong, β-strong and δ-arcs. The advantage of this type of classification is that it helps in understanding the basic structure of a fuzzy graph completely. We analyze the relation between strong paths and strongest paths in a fuzzy graph and obtain characterizations for fuzzy bridges, fuzzy trees and fuzzy cycles using the concept of α-strong, β-strong and δ-arcs. An arc of a fuzzy tree is α-strong if and only if it is an arc of its unique maximum spanning tree. Also we identify different types of arcs in complete fuzzy graphs.     

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.