Canonical decomposition for bipartite graphs, which was introduced by Fouquet, Giakoumakis, and Vanherpe (1999), is a decomposition scheme for bipartite graphs associated with modular decomposition. Weak-bisplit graphs are bipartite graphs totally decomposable (i.e., reducible to single vertices) by canonical decomposition. Canonical decomposition comprises series, parallel, and K+S decomposition. This paper studies a decomposition scheme comprising only parallel and K+S decomposition. We show that bipartite graphs totally decomposable by this decomposition are precisely P6-free chordal bipartite graphs. This characterization indicates that P6-free chordal bipartite graphs can be recognized in linear time using the recognition algorithm for weak-bisplit graphs presented by Giakoumakis and Vanherpe (2003).

We study the problem of transforming one (vertex) c-coloring of a graph into another one by changing only one vertex color assignment at a time, while at all times maintaining a c-coloring, where c denotes the number of colors. This decision problem is known to be PSPACE-complete even for bipartite graphs and any fixed constant c ≥ 4. In this paper, we study the problem from the viewpoint of graph classes. We first show that the problem remains PSPACE-complete for chordal graphs even if c is a fixed constant. We then demonstrate that, even when c is a part of input, the problem is solvable in polynomial time for several graph classes, such as k-trees with any integer k ≥ 1, split graphs, and trivially perfect graphs.

Given a graph G, a set of spanning trees of G are completely independent spanning trees (CISTs for short) if for any vertices x and y, the paths connecting them on these trees have neither vertex nor edge in common, except x and y. Hasunuma (2001, 2002) first introduced the concept of CISTs and conjectured that there are k CISTs in any 2k-connected graph. Later on, this conjecture was unfortunately disproved by Péterfalvi (2012). In this note, we show that Hasunuma's conjecture holds for graphs restricted in the class of 4-regular chordal rings CR(n,d), where both n and d are even integers.

Peer-to-peer (P2P) overlay networks are widely employed in distributed systems. The number of hops required by a node to locate an object is the fundamental search cost of a P2P network. Creating replicas can efficiently reduce the cost of object search, so how to deploy replicas to reduce the cost as much as possible is a critical problem of P2P networks. In the literature, most existing replica placement strategies arrange replicas at nodes near the one containing the considered object. In this paper, we formally demonstrate that for a complete Chord P2P network and many non-complete Chord ones, due to their deterministic structures, we can allocate replicas to nodes closest to the target in the identifier space to maximize the reduction in the total number of hops required by all nodes to reach a copy of the object during the search heading to the target node.

Two spanning trees T1,T2 of a graph G = (V,E) are independent if they are rooted at the same vertex, say r, and for each vertex v ∈ V, the path from r to v in T1 and the path from r to v in T2 have no common vertices and no common edges except for r and v. In general, spanning trees T1,T2,…,Tk of a graph G = (V,E) are independent if they are pairwise independent. A graph G = (V,E) is called a 2-chordal ring and denoted by CR(N,d1,d2), if V = {0,1,…,N-1} and E = {(u,v)|[v-u]N = 1 or [v-u]N = d1 or [v-u]N = d2, 2 ≤ d1 < d2 ≤ N/2}. CR(N,d1,N/2) is 5-connected if N ≥ 8 is even and d1 ≠ N/2-1. We give an algorithm to construct 5 independent spanning trees of CR(N,d1,N/2),N ≥ 8 is even and 2 ≤ d1 ≤ ⌈N/4⌉.

Let T1,T2,...,Tk be spanning trees in a graph G. If, for any two vertices u,v of G, the paths joining u and v on the k trees are mutually vertex-disjoint, then T1,T2,...,Tk are called completely independent spanning trees (CISTs for short) of G. The construction of CISTs can be applied in fault-tolerant broadcasting and secure message distribution on interconnection networks. Hasunuma (2001) first introduced the concept of CISTs and conjectured that there are k CISTs in any 2k-connected graph. Unfortunately, this conjecture was disproved by Péterfalvi recently. In this note, we give a necessary condition for k-connected k-regular graphs with ⌊k/2⌋ CISTs. Based on this condition, we provide more counterexamples for Hasunuma's conjecture. By contrast, we show that there are two CISTs in 4-regular chordal rings CR(N,d) with N=k(d-1)+j under the condition that k ≥ 4 is even and 0 ≤ j ≤ 4. In particular, the diameter of each constructed CIST is derived.

We consider a time-division multiple access (TDMA) transmission scheme in MIMO broadcast channels. To cope with the fairness issue in heterogeneous networks with slow fading, an opportunistic scheduling algorithm based on the channel eigen-direction is investigated. In the system with sparse users, the mismatch between a random beamforming vector and the principle eigenvector of the channel incurs a throughput penalty. To reduce such a throughput loss, a multiple random beams selection (MRBS) scheme exploiting chordal distances is proposed. Two feedback schemes (unquantized or quantized chordal distances) are considered. The closed-form throughput expressions of the proposed schemes are derived.

It is known that any chordal graph can be uniquely decomposed into simplicial components. Based on this fact, it is shown that for a given chordal graph, its automorphism group can be computed in O((c!n)O(1)) time, where c denotes the maximum size of simplicial components and n denotes the number of nodes. It is also shown that isomorphism of those chordal graphs can be decided within the same time bound. From the viewpoint of polynomial-time computability, our result strictly strengthens the previous ones respecting the clique number.

A chordal graph is a graph which contains no chordless cycle of at least four edges as an induced subgraph. The class of chordal graphs contains many famous graph classes such as trees, interval graphs, and split graphs, and is also a subclass of perfect graphs. In this paper, we address the problem of enumerating all labeled chordal graphs included in a given graph. We think of some variations of this problem. First we introduce an algorithm to enumerate all connected labeled chordal graphs in a complete graph of n vertices. Next, we extend the algorithm to an algorithm to enumerate all labeled chordal graphs in a n-vertices complete graph. Then, we show that we can use, with small changes, these algorithms to generate all (connected or not necessarily connected) labeled chordal graphs in arbitrary graph. All our algorithms are based on reverse search method, and time complexities to generate a chordal graph are O(1), and also O(1) delay. Additionally, we present an algorithm to generate every clique of a given chordal graph in constant time. Using these algorithms we obtain combinatorial Gray code like sequences for these graph structures in which the differences between two consecutive graphs are bounded by a constant size.

Routing efficiency is the critical issue when constructing peer-to-peer overlay. However, Chord has often been criticized on its careless of routing locality. A routing efficiency enhancement protocol on top of Chord is illustrated in this paper, which is called PChord. PChord aims to achieve better routing efficiency than Chord by exploiting proximity of the underlying network topology. The simulation shows that PChord has achieved lower RDP per message routing.

In this paper, we study the following problem: given two graphs G, H and an isomorphism φ between an induced subgraph of G and an induced subgraph of H, compute the number of isomorphisms between G and H that do not contradict φ. We show that this problem can be solved in O(((k+1)(k+1)!)2n3) time when the input graphs are restricted to chordal graphs with clique number at most k+1. To prove this, we first show that the tree model of a chordal graph can be uniquely constructed in O(n3) time except for the ordering of children of each node. Then, we show that the number of φ-isomorphisms between G and H can be efficiently computed by use of the tree model.

Coloring problem is a well-known combinatorial optimization problem of graphs. This paper considers H-coloring problems, which are coloring problems with restrictions such that some pairs of colors can not be used for adjacent vertices. The restriction of adjacent colors can be represented by a graph H, i.e., each vertex represents a color and each edge means that the two colors corresponding to the two end-vertices can be used for adjacent vertices. Especially, H-coloring problem with a complete graph H of order k is equivalent to the traditional k-coloring problem. This paper presents sufficient conditions such that H-coloring problem can be reduced to an H-coloring problem, where H is a subgraph of H. And it shows a hierarchy about classes of H-colorable graphs for any complement graph H of a cycle of order odd n 5.

This paper describes embeddings of chordal rings and pyramids into mesh-connected computers with multiple buses which have a bus on each row and each column, called MCCMBs. MCCMBs have two types of communication. The one is local communication, provided by local links, and the other is global communication, provided by buses. By efficiently combining the two types of communication, optimal or efficient embeddings are achieved. For a large set of chordal rings, optimal embeddings, whose expansion, load, dilation and congestion are 1, are given. For pyramids, an efficient embedding based on a two phase strategy is presented. The embedding balances dilation and congestion.

A multiple instruction stream-multiple data stream (MIMD) computer is a parallel computer consisting of a large number of identical processing elements. The essential feature that distinguishes one MIMD computer family from another is the interconnection network. In this paper, 2 representative types of interconnection networks are dealt with the chordal ring network and the mesh connected network. A family of regular graphs of degree 3, called chordal rings is presented as a possible candidate for the implementation of a distributed system and for fault-tolerant architectures. The symmetry of graphs makes it possible to determine message routing by using a simple distributed algorithm. Another candidate having the same property is the mesh connected networks. Arbitrary data permutations are generally accomplished by sorting. For certain classes of permutations, however, there exist algorithms that are more efficient than the best sorting algorithm. One such class is the bit permute complement (BPC) class of permutations. The class of BPC permutations includes many of the frequently occurring permutations such as bit reversal, bit shuffle, bit complement, matrix transpose, etc. In this paper, we evaluate the abilities of the above networks to realize BPC permutations. In this paper, we, first, develop algorithms required 2 token storage registers in each node to realize an arbitrary BPC permutaion in both chordal ring networks and mesh connected networks. We next evaluate the ability to realize BPC permutations in these networks of an arbitrary size by estimating the number of required routing steps.

A chordal ring network is a processor network on which n processors are arranged to a ring with additional chords. We study a distributed leader election algorithm on chordal ring networks and present trade-offs between the message complexity and the number of chords at each processor and between the message complexity and the length of chords as follows:For every d(1dlog* n1) there exists a chordal ring network with d chords at each processor on which the message complexity for leader election is O(n(log(d1)nlog* n)).For every d(1dlog* n1) there exists a chordal ring network with log(d1)nd1 chords at each processor on which the message complexity for leader election is O(dn).For every m(2mn/2) there exists a chordal ring network whose chords have at most length m such that the message complexity for leader election is O((n/m)log n).