Broadcasting is the process of information dissemination in communication networks (modelled as graphs) whereby a message originating at one vertex becomes known to all members under the constraint that each call requires one unit of time and at every step any member can call at most one of its neighbours. A broadcast graph on n vertices is a network in which message can be broadcast in the minimum possible (= [log(2)n]) time regardless of the originator. Broadcast graphs having the smallest number of edges are called minimum broadcast graphs, and are subjects of intensive study. On the other hand, in Shastri (1995) we have considered how quickly broadcasting can be done in trees. In this paper, we study how the number of edges in a minimum broadcast graphs decrease, as we allow addition time over [log(2)n], until we get a tree. In particular, the sparsest possible time-relaxed broadcast graphs are constructed for small n(less than or equal to 15) and very sparse time-relaxed broadcast graphs are given for larger n(less than or equal to 65). General constructions are also provided putting bounds which hold for all n. Some of these constructions make use of the techniques developed in Bermond et al. (1995, 1992) and Chau and Liestman (1985). (C) 1998 Elsevier Science B.V. All rights reserved. [References: 16]
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