S. Louis Hakimi

On computing a conditional edge-connectivity of a graph

Abstract The conditional edge-connectivity λ ( G : P ) of a graph G ( V , E ) has been defined by Harary as the minimum cardinality | S | of a set S of edges such that G – S is disconnected and every component of G – S has the given graph property P . I n this article we present lower and upper bounds for λ( G : P ) when P is defined as follows: A graph H satisfies property P if it contains more than one vertex. We then present a polynomial-time algorithm for the computation of λ( G : P ). A new generalization of the notion of connectivity is also given.

Market and locational equilibrium for two competitors

We consider a two-stage location and allocation game involving two competing firms. The firms first select the location of their facility on a network. Then the firms optimally select the quantities each wishes to supply to the markets, which are located at the vertices of the network. The criterion for optimality for each firm is maximizing its profit, which is the total revenue minus the production and transportation costs. Under reasonable assumptions regarding the revenue, the production cost and the transportation cost functions, we show that there is a Nash equilibrium for the quantities offered at the markets by each firm. Furthermore, if the quantities supplied at the equilibrium by each firm at each market are positive, then there is also a Nash locational equilibrium, i.e., no firm finds it advantageous to change its location.

The Voronoi partition of a network and its implications in location theory

Given a network N(V, E) and a set of points Xp = {x1, …, xp} on N, we first present an algorithm for computing the Voronoi partition of N(V, E) into territories T(x1), …, T(xp). After describing two ways to measure the “size” of a territory, we introduce and discuss the more challenging problem of selecting Xp so that the maximum size among the resulting territories is as small as possible. For one especially natural way to measure the size of a territory, we show that this latter problem is NP-complete when p is part of the input, but that the problem can be solved in polynomial time for any fixed p. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Disjoint Rooted Spanning Trees with Small Depths in deBruijn and Kautz Graphs

The problem of broadcasting long messages on store-and-forward communication networks, where a processor (node) can send and receive messages simultaneously to and from all its neighbors, was studied by Bermond and Fraigniaud. In such networks, the delays encountered by a message from a node

Data transfers in networks

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Scheduling file transfers for trees and odd cycles

to all other nodes over a broadcast spanning tree is directly proportional to the length of the paths in the tree over which the message is sent. Furthermore, the speed of the broadcast can be improved by the segmentation of the message at

Orienting graphs to optimize reachability

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On a general network location-production-allocation problem

into equal-length segments and then the broadcast of these segments over arc-disjoint broadcast spanning trees simultaneously. These observations lead Bermond and Fraigniaud to look for the maximum number of arc-disjoint spanning trees in a deBruijn network rooted at an arbitrary node with small depths. This paper improves and extends the results of the above authors.

Locating replicas of a database on a network

The scheduling of the transfer of backlogged data in a network to minimize the finishing time is studied. The most complete treatment (of a version) of the problem is due to Gopal, Bongiovanni, Bonucelli, Tang, and Wong, who attacked the problem using the Birkhoff-von Neumann theorem. However, these authors do not provide a complexity analysis of their algorithm.In this paper we solve the version of these authors as well as a more difficult version of this scheduling problem by formulating them as a continuous form of the Hakimi-Kariv-de Werra generalization of the edge-coloring problem in bipartite graphs. This leads to polynomial time algorithms for these problems. Furthermore, our solution of the previously solved version has the desirable feature of having a tighter bound for the number of “communication modes” than the solution of the above authors.In the above scheduling problem, there may be a time associated with changing from one set of simultaneous data transfers (i.e., a communication mode) to another. It is shown that if the overall finishing time of our schedule includes these times, then even very simple instances of our problem become NP-hard. However, approximation algorithms are presented which produce solutions whose finishing times are at most twice the optimal.Finally, in the above scheduling problem the interruption (or pre-emption) of the performance of each task is permitted. Essentially, the same problem when pre-emption is not permitted was studied by Coffman, Garey, Johnson, and LaPaugh. The relation between the two problems are explored.

Improved bounds for the chromatic index of graphs and multigraphs

The scheduling of file transfers in networks to minimize the overall finishing time was studied by Coffman, et al. where the schedule does not permit interruption and each communication module can be used as a transmitter and as a receiver. They first presented complexity results under various conditions. Then they showed that the general problem is NP-complete and provided approximation algorithms. This paper first presents more efficient approximation algorithms with better performances than the above authors’ algorithms for the cases of trees and multitrees. Furthermore, there are simple distributed implementations of our approximation algorithms. Then this paper provides a polynomial time algorithm for finding an optimum schedule for an odd cycle, whose complexity was left as an open question by the above authors.