Jou-Ming Chang

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.

Induced matchings in asteroidal triple-free graphs

An induced matching M of a graph G is a set of pairwise nonadjacent edges such that no two edges of M are joined by an edge in G. The problem of finding a maximum induced matching is known to be NP-hard even for bipartite graphs of maximum degree four. In this paper, we study the maximum induced matching problem on classes of graphs related to AT-free graphs. We first define a wider class of graphs called the line-asteroidal triple-free (LAT-free) graphs which contains AT-free graphs as a subclass. By examining the square of line graph of LAT-free graphs, we give a characterization of them and apply this for showing that the maximum induced matching problem and a generalization, called the maximum δ-separated matching problem, on LAT-free graphs can be solved in polynomial time. In fact, our result can be extended to the classes of graphs with bounded asteroidal index. Next, we propose a linear-time algorithm for finding a maximum induced matching in a bipartite permutation (bipartite AT-free) graph using the greedy approach. Moreover, we show that using the same technique the minimum chain subgraph cover problem on bipartite permutation graphs can be solved with the same time complexity.

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).

Feedback vertex sets in star graphs

In a graph G = (V, E), a subset F ⊂ V(G) is a feedback vertex set of G if the subgraph induced by V(G) \ F is acyclic. In this paper, we propose an algorithm for finding a small feedback vertex set of a star graph. Indeed, our algorithm can derive an upper bound to the size of the feedback vertex set for star graphs. Also by applying the properties of regular graphs, a lower bound can easily be achieved for star graphs.

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.

A linear time algorithm for binary tree sequences transformation using left-arm and right-arm rotations

In this paper, we consider a transformation on binary trees using new types of rotations. Each of the newly proposed rotations is permitted only at nodes on the left-arm or the fight-arm of a tree. Consequently, we develop a linear time algorithm with at most n - 1 rotations for converting weight sequences between any two binary trees. In particular, from an analysis of aggregate method for a sequence of rotations, each rotation of the proposed algorithm can be performed in a constant amortized time. Next, we show that a specific directed rooted tree called rotation tree can be constructed using one of the new type rotations. As a by-product, a naive algorithm for enumerating weight sequences of binary trees in lexicographic order can be implemented by traversing the rotation tree.

LexBFS-Ordering in Asteroidal Triple-Free Graphs

In this paper, we study the metric property of LexBFS-ordering on AT-free graphs. Based on a 2-sweep LexBFS algorithm, we show that every AT-free graph admits a vertex ordering, called the strong 2-cocomparability ordering, that for any three vertices u ≺ v ≺ w in the ordering, if d(u, w) ≤ 2 then d(u, v) = 1 or d(v, w) ≤ 2. As an application of this ordering, we provide a simple linear time recognition algorithm for bipartite permutation graphs, which form a subclass of AT-free graphs.

Loopless Generation of Non-regular Trees with a Prescribed Branching Sequence1

An ordered tree is called a non-regular tree with a prescribed branching sequence (or non-regular tree for short) if its internal nodes have a prespecified degree sequence in preorder list. We define a concise representation, called right distance sequences to describe such trees. A coding tree helps us to systematically investigate the structural representation of non-regular trees. Consequently, we present a loopless algorithm to generate Gray-codes of non-regular trees using right distance sequences.

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.