205001978
A Linear Time Algorithm to Compute a Dominating Path in an AT-Free Graph
Abstract An independent set { x , y , z } is called an asteroidal triple if between any pair in the triple there exists a path that avoids the neighborhood of the third. A graph is referred to as AT-free if it does not contain an asteroidal triple. We present a simple linear-time algorithm to compute a dominating path in a connected AT-free graph.
204991386
Quasi-brittle graphs, a new class of perfectly orderable graphs
Abstract A graph G is quasi-brittle if every induced subgraph H of G contains a vertex which is incident to no edge extending symmetrically to a chordless path with three edges in either H or its complement H¯ . The quasi-brittle graphs turn out to be a natural generalization of the well-known class of brittle graphs. We propose to show that the quasi-brittle graphs are perfectly orderable in the sense of Chvatal: there exists a linear order a , b , c , d and edges ab , bc , cd has a b and d c .
7817544
P-Components and the Homogeneous Decomposition of Graphs
In this paper we introduce and investigate the notion of p-connectedness. As it turns out, this concept leads naturally to a unique tree representation for arbitrary graphs: the leaves of this tree are the p-connected components along with weak vertices, that is, vertices of the graph that belong to no p-connected component. We then show how to refine this decomposition to obtain a new decomposition that extends the well-known modular decomposition.
37000292
Wings and perfect graphs
Abstract An edge uv of a graph G is called a wing if there exists a chordless path with vertices u , v , x , y and edges uv , vx , xy . The wing-graph W ( G ) of a graph G is a graph having the same vertex set as G ; uv is an edge in W ( G ) if and only if uv is a wing in G . A graph G is saturated if G is isomorphic to W ( G ). A star-cutset in a graph G is a non-empty set of vertices such that G — C is disconnected and some vertex in C is adjacent to all the remaining vertices in C . V. Chvatal proposed to call a graph unbreakable if neither G nor its complement contain a star-cutset. We establish several properties of unbreakable graphs using the notions of wings and saturation. In particular, we obtain seven equivalent versions of the Strong Perfect Graph Conjecture.
5840834
Asteroidal Triple-Free Graphs
Many families of perfect graphs such as interval graphs, permutation graphs, trapezoid graphs and cocomparability graphs demonstrate a type of linear ordering of their vertex sets. These graphs are all subfamilies of a class of graphs called the asteroidal triple-free graphs. (An independent triple {x, y, z} is called an asteroidal triple (AT, for short) if between any pair in the triple there exists a path that avoids the neighbourhood of the third vertex.) In this paper we argue that the property of being AT-free is what is enforcing the linear ordering of the vertex sets. To justify this claim, we present various structural properties and characterizations of AT-free graphs.
32558867
Linear Time Algorithms for Dominating Pairs in Asteroidal Triple-free Graphs
An independent set of three of vertices is called an asteroidal triple if between each, pair in the triple there exists a path that avoids the neighbourhood of the third. A graph is asteroidal triple-free (AT-free, for short) if it contains no asteroidal triple. The motivation for this investigation is provided, in part, by the fact that AT-free graphs offer a common generalization of interval, permutation, trapezoid, and cocomparability graphs. Previously, the authors have given an existential proof of the fact that every connected AT-free graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. The main contribution of this paper is a constructive proof of the existence of dominating pairs in connected AT-free graphs. The resulting simple algorithm, based on the well-known Lexicographic Breadth-First Search, can be implemented to run in time linear in the size of the input, whereas the best algorithm previously known for this problem has complexity O(¦V¦3) for input graph G=(V, E). In addition, we indicate how our algorithm can be extended to find, in time linear in the size of the input, all dominating pairs in a connected AT-free graph with diameter greater than three. A remarkable feature of the extended algorithm is that, even though there may be O(¦V¦2) dominating pairs, the algorithm can compute and represent them in linear time.
5533650
The hierarchical cliques interconnection network
The fully connected network possesses extremely good topological, fault-tolerant, and embedding properties. However, due to its high degree, the fully connected network has not been an attractive candidate for building parallel computers. On the other hand, tree-based networks are popular as parallel computer networks, even though they suffer from poor fault-tolerance and embedding properties. The hierarchical cliques interconnection network described in this paper incorporates positive features of the fully connected network and the tree network. In other words, the hierarchical cliques possess such desirable properties as low diameter, low degree, self routing, versatile embedding, good fault-tolerance and strong resilience. Hierarchical cliques can efficiently embed most important networks and possess a scalable, modular structure. Further, by combining hierarchical cliques with fat trees congestion in the upper levels can be alleviated.
31459718
Sequential and Parallel Meta-Heuristics for Solving the Single Row Routing Problem
The design of multi-layer printed circuit boards is vital in the construction of complex electronic systems. Wire routing is a crucial step in the overall design process, which can be decomposed into a number of single row routing (SRR) problems. This paper proposes an approach to solve the SRR problem based on parallel meta-heuristics. The development of this technique involves the design of an encoding strategy that allows all possible routings to be uniquely represented and the derivation of cost functions that maximizes the quality of the developed solutions. Further, parallelization of the proposed approach is attempted to improve the computational efficiency. The different stages of the development are backed by experiments to show the pros and cons of the sequential and parallel implementations.
9333297
A two-zone hybrid routing protocol for mobile ad hoc networks
Recently, an elegant routing protocol, the zone routing protocol (ZRP), was proposed to provide a hybrid routing framework that is locally proactive and globally reactive, with the goal of minimizing the sum of the proactive and reactive control overhead. The key idea of ZRP is that each node proactively advertises its link state over a fixed number of hops, called the zone radius. These local advertisements provide each node with an updated view of its routing zone - the collection of all nodes and links that are reachable within the zone radius. The nodes on the boundary of the routing zone are called peripheral nodes and play an important role in the reactive zone-based route discovery. The main contribution of this work is to propose a novel hybrid routing protocol - the two-zone routing protocol (TZRP) - as a nontrivial extension of ZRP. In contrast with the original ZRP where a single zone serves a dual purpose, TZRP aims to decouple the protocol's ability to adapt to traffic characteristics from its ability to adapt to mobility. In support of this goal, in TZRP each node maintains two zones: a crisp zone and a fuzzy zone. By adjusting the sizes of these two zones independently, a lower total routing control overhead can be achieved. Extensive simulation results show that TZRP is a general MANET routing framework that can balance the trade offs between various routing control overheads more effectively than ZRP in a wide range of network conditions.
2805859
A time-optimal solution for the path cover problem on cographs
We show that the notoriously difficult problem of finding and reporting the smallest number of vertex-disjoint paths that cover the vertices of a graph can be solved time- and work-optimally for cographs. Our result implies that for this class of graphs the task of finding a Hamiltonian path can be solved time- and work-optimally in parallel.It was open for more than 10 years to find a time- and work-optimal parallel solution for this important problem. Our contribution is to offer an optimal solution to this important problem. We begin by showing that any algorithm that solves an instance of size n of the problem must take ?(logn) time on the CREW, even if an infinite number of processors are available. We then go on to show that this time lower bound is tight by devising an EREW algorithm that, given an n-vertex cograph G represented by its cotree, finds and reports all the paths in a minimum path cover in O(logn) time using n/logn processors.
14933759
Classifying matrices separating rows and columns
The classification problem transforms a set of N numbers in such a way that none of the first N/2 numbers exceeds any of the last N/2 numbers. A comparator network that solves the classification problem on a set of r numbers is commonly called an r-classifier. We show how the well-known Leighton's Columnsort algorithm can be modified to solve the classification problem of N=rs numbers, with 1 /spl les/ s /spl les/ r, using an r-classifier instead of an r-sorting network. Overall, the r-classifier is used O(s) times, namely, the same number of times that Columnsort applies an r-sorter. A hardware implementation is proposed that runs in optimal O(s+logr) time and uses an O(rlogr(s + logr)) work. The implementation shows that, when N= rlogr, there is a classifier network solving the classification problem on N numbers in the same O(logr) time and using the same O(rlogr) comparators as an r-classifier, thus saying a logr factor in the number of comparators over an (rlogr)-classifier.