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This paper is concerned with a particular family of regular 4-connected graphs, called chordal rings. Chordal rings are a variation of ring networks. By adding two extra links (or chords) at each vertex in a ring network, the reliability and fault-tolerance of the network are enhanced. Two spanning trees on a graph are said to be independent if they are rooted at the same vertex, say, r, and for each vertex vner, the two paths from r to v, one path in each tree, are internally disjoint. A set of spanning trees on a given graph is said to be independent if they are pairwise independent. Iwasaki et al. (1999) proposed a linear time algorithm for finding four independent spanning trees on a chordal ring. In this paper, we give a new linear time algorithm to generate four independent spanning trees with a reduced height in each tree. Moreover, a complete analysis of our improvements on the heights of independent spanning trees is also provided     

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.     

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.     

Compact Routing on Chordal Rings of Degree 4

A chordal ring G(n;c) of degree 4 is a ring of n nodes with chords connecting each vertex i to the vertex (i + c) mod n . In this paper we investigate compact routing schemes on such networks. We show an optimal boolean routing scheme for any such network that requires O( log n) bits of storage at each node, and O(1) time to compute a shortest path to any destination. This improves on the results of [16] which gives a linear time algorithm for such networks and [6] where efficient routing schemes for certain fixed values of c were developed. Further, we show several bounds on interval routing schemes for such networks. We show that while every chordal ring has an optimal interval routing scheme with at most intervals on any edge, there exist chordal rings for which any optimal interval routing scheme that labels the vertices around the ring in the graph requires intervals on some edges. Additionally, there are chordal rings which admit no optimal one-interval routing schemes, regardless of the vertex labeling. We also consider interval routing schemes under relaxed requirements for the lengths of paths. Received September 5, 1997; revised December 1, 1997.     

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.     

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.     

k-tuple domination in graphs

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the k-tuple domination problem is to find a minimum sized vertex subset in a graph such that every vertex in the graph is dominated by at least k vertices in this set. The current paper studies k-tuple domination in graphs from an algorithmic point of view. In particular, we give a linear-time algorithm for the k-tuple domination problem in strongly chordal graphs, which is a subclass of chordal graphs and includes trees, block graphs, interval graphs and directed path graphs. We also prove that the k-tuple domination problem is NP-complete for split graphs (a subclass of chordal graphs) and for bipartite graphs.     

基于CREW遍历图的一种并行算法

针对串行算法模型下基于顶点遍历图的情况,提出了一种在CREWPRAM并行模型下遍历无向图的算法。该算法是找出无向图的一棵最短路径生成树,由向上和向下两条有向边替换最短路径生成树的每条边形成欧拉回路,运用欧拉回路技术计算前缀和,前缀和所对应的顶点即为遍历无向图的顺序。得出了该算法时间复杂度为O（n＋logn）的结论。     

Tree spanners on chordal graphs: complexity and algorithms

A tree t-spanner T in a graph G is a spanning tree of G such that the distance in T between every pair of vertices is at most t times their distance in G. The T t-S problem asks whether a graph admits a tree t-spanner, given t. We substantially strengthen the hardness result of Cai and Corneil (SIAM J. Discrete Math. 8 (1995) 359–387) by showing that, for any t4, T t-S is NP-complete even on chordal graphs of diameter at most t+1 (if t is even), respectively, at most t+2 (if t is odd). Then we point out that every chordal graph of diameter at most t−1 (respectively, t−2) admits a tree t-spanner whenever t2 is even (respectively, t3 is odd), and such a tree spanner can be constructed in linear time.

The complexity status of T 3-S still remains open for chordal graphs, even on the subclass of undirected path graphs that are strongly chordal as well. For other important subclasses of chordal graphs, such as very strongly chordal graphs (containing all interval graphs), 1-split graphs (containing all split graphs) and chordal graphs of diameter at most 2, we are able to decide T 3-S efficiently.     

The algorithmic complexity of mixed domination in graphs

Let G=(V,E) be a simple graph with vertex set V and edge set E. A subset W⊆V∪E is a mixed dominating set if every element x∈(V∪E)?W is either adjacent or incident to an element of W. The mixed domination problem is to find a minimum mixed dominating set of G. In this paper we first prove that a connected graph is a tree if and only if its total graph is strongly chordal, and thus we obtain a polynomial-time algorithm for this problem in trees. Further we design another linear-time labeling algorithm for this problem in trees. At the end of the paper, we show that the mixed domination problem is NP-complete even when restricted to split graphs, a subclass of chordal graphs.     

A linear algorithm for the number of degree constrained subforests of a tree

Recent work of Farrell is concerned with determining the total number of ways in which one can cover the vertices of a tree T with vertex disjoint paths. Such path covers are spanning subforests in which each vertex is restricted to have degree at most two. If b: V(T)→N is a positive integer-valued function, then finding a b-matching is equivalent to finding a spanning subgraph in which the degree of each vertex v is at most b(v). A linear-time algorithm for determining the number of b-matching in a tree is presented.     

Augmented ring networks

We study four augmentations of ring networks which are intended to enhance a ring's efficiency as a communication medium significantly, while increasing its structural complexity only modestly. Chordal rings add “shortcut” edges, which can be viewed as chords, to the ring. Express rings are chordal rings whose chords are routed outside the ring. Multirings append subsidiary rings to edges of a ring and, recursively, to edges of appended subrings. Hierarchical ring networks (HRN's) append subsidiary rings to nodes of a ring and, recursively, to nodes of appended subrings. We show that these four modes of augmentation are very closely related: 1) Planar chordal rings, planar express rings, and multirings are topologically equivalent families of networks with the “cutwidth” of an express ring translating into the “tree depth” of its isomorphic multiring and vice versa. 2) Every depth-d HRN is a spanning subgraph of a depth-(2d-1) multiring. 3) Every depth-d multiring ℳ can be embedded into a d-dimensional mesh with dilation 3 in such a way that some node of ℳ resides at a corner of the mesh. 4) Every depth-d HRN ℋ can be embedded into a d-dimensional mesh with dilation 2 in such a way that some node of ℋ resides at a corner of the mesh. In addition to demonstrating that these four augmented ring networks are grid graphs, our embedding results afford us close bounds on how much decrease in diameter is achievable for a given increase in structural complexity for the networks. Specifically, we derive upper and lower bounds on the optimal diameters of N-node depth-d multirings and HRN's that are asymptotically tight for large N and d     

Decision tree complexity of graph properties with dimension at most 5

A graph property is a set of graphs such that if the set contains some graph G then it also contains each isomorphic copy of G(with the same vertex set).A graph propoerty P on n vertices is said to be elusive,if every decision tree algorithm recognizing P must examine all n(n-1)/2 paris of vertices in the worst case.Karp conjectured that every nontrivial monotone graph property is elusive,In this paper,this conjecture is proved for some cases,Especially,it is shown that if the abstract simplicial complex of a nontrivial monotone graph property P has dimension not exceeding 5, then P is elusive.     

A graph approach to generate all possible regression submodels

A regression graph to enumerate and evaluate all possible subset regression models is introduced. The graph is a generalization of a regression tree. All the spanning trees of the graph are minimum spanning trees and provide an optimal computational procedure for generating all possible submodels. Each minimum spanning tree has a different structure and characteristics. An adaptation of a branch-and-bound algorithm which computes the best-subset models using the regression graph framework is proposed. Experimental results and comparison with an existing method based on a regression tree are presented and discussed.     

An O(log n) algorithm for parallel update of minimum spanning trees

Parallel algorithms are presented for updating a minimum spanning tree when the cost of an edge changes or when a new node is inserted in the underlying graph. Our model of computation is a parallel random access machine which allows simultaneous reads but prohibits simultaneous writes into the same memory location. The algorithms described in this paper for updating a minimum spanning tree require O(log n) time and O(n²) processors. These algorithms are efficient when compared to previously known algorithms for initial construction of a minimum spanning tree that require O(log²n) time and use O(n²) processors. The two main contributions of this paper are: (i) usage of an inverted tree for updating minimum spanning trees, and (ii) discovery of an interesting property of minimum spanning trees that we exploit to develop a novel algorithm for vertex insertion update.     

A Linear-Time Algorithm for Finding Locally Connected Spanning Trees on Circular-Arc Graphs

Suppose that T is a spanning tree of a graph G. T is called a locally connected spanning tree of G if for every vertex of T, the set of all its neighbors in T induces a connected subgraph of G. In this paper, given an intersection model of a circular-arc graph, an O(n)-time algorithm is proposed that can determine whether the circular-arc graph contains a locally connected spanning tree or not, and produce one if it exists.     

Degree constrained minimum spanning tree problem: a learning automata approach

Degree-constrained minimum spanning tree problem is an NP-hard bicriteria combinatorial optimization problem seeking for the minimum weight spanning tree subject to an additional degree constraint on graph vertices. Due to the NP-hardness of the problem, heuristics are more promising approaches to find a near optimal solution in a reasonable time. This paper proposes a decentralized learning automata-based heuristic called LACT for approximating the DCMST problem. LACT is an iterative algorithm, and at each iteration a degree-constrained spanning tree is randomly constructed. Each vertex selects one of its incident edges and rewards it if its weight is not greater than the minimum weight seen so far and penalizes it otherwise. Therefore, the vertices learn how to locally connect them to the degree-constrained spanning tree through the minimum weight edge subject to the degree constraint. Based on the martingale theorem, the convergence of the proposed algorithm to the optimal solution is proved. Several simulation experiments are performed to examine the performance of the proposed algorithm on well-known Euclidean and non-Euclidean hard-to-solve problem instances. The obtained results are compared with those of best-known algorithms in terms of the solution quality and running time. From the results, it is observed that the proposed algorithm significantly outperforms the existing method.     

基于“最小差值法”的网络模型耗费代价研究

最小生成树算法是数据结构中,求网络模型耗费代价最优解的一个重要工具。现实生活中的连通网络模型复杂而多变,有时还需兼顾其它的目标,一棵最小生成树不足以解决问题,因此找出所有的最小生成树是很有必要的,在此提出一种新的寻找所有最小生成树的算法--最小差值法。无向连通图网络通过去掉连枝生成最小生成树,一个连枝加入最小生成树形成一个圈。这种算法是在一个圈中,用连枝的权与其它树枝的权分别作差,求最小差值。由最小差值是否为零,判断原有的最小生成树能否通过换进换出边,生成新的最小生成树。该算法能够有规律、高效率的寻找出所有的最小生成树。在找出的所有最小生成树方案中,选择符合实时情况的最小生成树方案,该方案即为网络耗费代价的最优解。     

Vertex rankings of chordal graphs and weighted trees

In this paper we consider the vertex ranking problem of weighted trees. We show that this problem is strongly NP-hard. We also give a polynomial-time reduction from the problem of vertex ranking of weighted trees to the vertex ranking of (simple) chordal graphs, which proves that the latter problem is NP-hard. In this way we solve an open problem of Aspvall and Heggernes. We use this reduction and the algorithm of Bodlaender et al.'s for vertex ranking of partial k-trees to give an exact polynomial-time algorithm for vertex ranking of a tree with bounded and integer valued weight functions. This algorithm serves as a procedure in designing a PTAS for weighted vertex ranking problem of trees with bounded weight functions.     

Constructing dual-CISTs of pancake graphs and performance assessment of protection routings on some Cayley networks

The Journal of Supercomputing - For a connected graph $$G=(V,E)$$ , two spanning trees $$T_1$$ and $$T_2$$ of G are said to be a pair of completely independent spanning trees (or a dual-CIST for...