Broadcasting in generalized chordal rings

Abstract

AbstractBroadcasting is an information dissemination process in which a message originating at one node of a communication network (modeled as an undirected graph) is sent to all other nodes by means of calls involving two nodes at a time, with each node participating in at most one call at any time. We are interested in efficient broadcasting in a class of cubic graphs known as generalized chordal rings. These graphs have been found useful for having a small diameter D, among graphs with a given number of vertices and maximum degree. We show that the minimum broadcast time in any generalized chordal ring is D, D + 1, or D + 2. For the generalized chordal rings of diameter D which have the greatest (or greatest‐known) number of nodes, we then evaluate exactly the minimum broadcast time. It turns out to be D + 1 when D is even and D + 2 when D is odd. For these purposes, we review the construction of these extremal generalized chodal rings. We also review the optimal broadcast schemes for infinite triangular grids, which we use to prove our bounds. Finally, we ask for the maximum number of nodes that can be informed by a broadcast in time t in any generalized chordal ring. We answer this completely for even t and almost completely for odd t. We use a geometric approach, based on plane tessellations. © 2003 Wiley Periodicals, Inc.