Jinn-Shyong Yang

Reducing the Height of Independent Spanning Trees in Chordal Rings

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 v \neq r, 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. [CHECK END OF SENTENCE] 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.

Independent Spanning Trees on Multidimensional Torus Networks

Two spanning trees rooted at vertex r in a graph G are called independent spanning trees (ISTs) if for each vertex v in G, vner, the paths from vertex v to vertex r in these two trees are internally distinct. If the connectivity of G is k, the IST problem is to construct k ISTs rooted at each vertex. The IST problem has found applications in fault-tolerant broadcasting, but it is still open for general graphs with connectivity greater than four. In this paper, we shall propose a very simple algorithm for solving the IST problem on multidimensional torus networks. In our algorithm, every vertex can determine its parent for a specific independent spanning tree only depending on its own label. Thus, our algorithm can also be implemented in parallel systems or distributed systems very easily.

On the independent spanning trees of recursive circulant graphs G(cdm,d) with d>2

Two spanning trees of a graph G are said to be independent if they are rooted at the same vertex r, and for each vertex v r in G, the two different paths from v to r, one path in each tree, are internally disjoint. A set of spanning trees of G is independent if they are pairwise independent. The construction of multiple independent spanning trees has many applications in network communication. For instance, it is useful for fault-tolerant broadcasting and secure message distribution. A recursive circulant graph G(N,d) has N=cd^m vertices labeled from 0 to N-1, where d>=2, m>=1, and 1=2, where the number of independent spanning trees matches the connectivity of G(cd^m,d).

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.

CONSTRUCTING MULTIPLE INDEPENDENT SPANNING TREES ON RECURSIVE CIRCULANT GRAPHS G(2 m ,2) ∗

A recursive circulant graph G(N,d) has N = cdm vertices labeled from 0 to N - 1, where d ⩾ 2, m ⩾ 1, and 1 ⩽ c < d, and two vertices x,y ∈ G(N,d) are adjacent if and only if there is an integer k with 0 ⩽ k ⩽ ⌈logd N⌉ - 1 such that x ± dk ≡ y (mod N). With the aid of recursive structure, such class of graphs has many attractive features and was considered as a topology of interconnection networks for computing systems. The design of multiple independent spanning trees (ISTs) has many applications in network communication. For instance, it is useful for fault-tolerant broadcasting and secure message distribution. In the previous work of Yang et al. (2009), we provided a constructing scheme to build k ISTs on G(cdm,d) with d ⩾ 3, where k is the connectivity of G(cdm,d). However, the proposed constructing rules cannot be applied to the case of d = 2. For the integrity of solving the IST problem on recursive circulant graphs, this paper deals with the case of G(2m,2) using a set of different constructing rules. Especially, we show that the heights of ISTs for G(2m,2) are lower than the known optimal construction of hypercubes with the same number of vertices.

Parallel Construction of Independent Spanning Trees on Enhanced Hypercubes

The use of multiple independent spanning trees (ISTs) for data broadcasting in networks provides a number of advantages, including the increase of fault-tolerance, bandwidth and security. Thus, the designs of multiple ISTs on several classes of networks have been widely investigated. In this paper, we give an algorithm to construct ISTs on enhanced hypercubes Qn,k, which contain folded hypercubes as a subclass. Moreover, we show that these ISTs are near optimal for heights and path lengths. Let D(Qn,k) denote the diameter of Qn,k. If n - k is odd or n - k ∈ {2; n}, we show that all the heights of ISTs are equal to D(Qn,k) + 1, and thus are optimal. Otherwise, we show that each path from a node to the root in a spanning tree has length at most D(Qn,k) + 2. In particular, no more than 2.15 percent of nodes have the maximum path length. As a by-product, we improve the upper bound of wide diameter (respectively, fault diameter) of Qn,k from these path lengths.

Optimal Independent Spanning Trees on Cartesian Product of Hybrid Graphs

A set of k spanning trees rooted at the same vertex r in a graph G are called independent spanning trees (ISTs) if for any vertex x 6= r, the k paths from v to r, one path in each tree, are internally disjoint. The design of ISTs on graphs has applications to fault-tolerant broadcasting and secure message distribution in networks. It was conjectured that for any k-connected graph there exist k ISTs rooted at any vertex of the graph. The conjecture has been proved true for k-connected graphs with k 6 4, and remains open otherwise. In this paper, we deal with the problem of constructing ISTs on Cartesian product of a sequence of hybrid graphs including cycles and complete graphs. Consequently, this result generalizes a number of previous works. Keyword: independent spanning trees; Cartesian product; fault-tolerant broadcasting;

Independent spanning trees on folded hyper‐stars

Fault‐tolerant broadcasting and secure message distribution are important issues for numerous applications in networks. It is a common idea to design multiple independent spanning trees (ISTs) as a broadcasting scheme or a distribution protocol for receiving high levels of fault‐tolerance and security. Recently, hyper‐stars were introduced as a competitive model of interconnection network for both hypercubes and star graphs. The class of folded hyper‐stars is a strengthened variation of hyper‐stars obtained by adding additional links to connect complemented nodes. Both hyper‐stars and folded hyper‐stars have been shown to have lower network cost (measured by the product of degree and diameter) than hypercubes, folded hypercubes, and other variants. In this article, we propose an algorithm to construct k + 1 ISTs on a regular folded hyper‐star FHS (2k,k), where the number of ISTs matches the connectivity of FHS(2k,k). In particular, for k > 4, the constructed k ISTs have height 2 k − 2, and the other one has height k + 1.

Completely Independent Spanning Trees on Complete Graphs, Complete Bipartite Graphs and Complete Tripartite Graphs

Let T 1, T 2,…, T k be spanning trees in a graph G. If for any two vertices x, y of G, the paths from x to y in T 1, T 2,…, T k are vertex-disjoint except end vertices x and y, then T 1, T 2,…, T k are called completely independent spanning trees in G. In 2001, Hasunuma gave a conjecture that there are k completely independent spanning trees in any 2k-connected graph. Peterfalvi disproved the conjecture in 2012. In this paper, we shall prove that there are \(\lfloor\frac{n}{2}\rfloor\) completely independent spanning trees in a complete graph with \(n ( \geqslant 4)\) vertices. Then, we prove that there are \(\lfloor\frac{n}{2}\rfloor\) completely independent spanning trees in a complete bipartite graph K m,n where \(m\geqslant n\geqslant 4\). Next, we also prove that there are \(\lfloor\frac{n_1+n_2}{2}\rfloor\) completely independent spanning trees in a complete tripartite graph \(K_{n_3,n_2,n_1}\) where \(n_3\geqslant n_2\geqslant n_1\) and \(n_1+n_2\geqslant 4\). As a result, the Hasunuma’s conjecture holds for complete graphs and complete m-partite graphs where m ∈ {2,3}.

Independent Spanning Trees on Folded Hypercubes

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 they are rooted at the same node without sharing any common edge. A folded hypercube is a strengthened variation of hypercube obtained by adding additional links between nodes that are Hamming distance furthest apart. Ho [C.-T. Ho, Full bandwidth communications on folded hypercubes, in Proc. 1990 International conference on Parallel Processing, vol. 1, 1990, pp. 267-280] presented an algorithm for constructing n+1 edge-disjoint spanning trees in an n-dimensional folded hypercube. In this paper, we prove that indeed all spanning trees constructed by this algorithm are independent, i.e., any two spanning trees have the same root, say r, and for any other node