Discrete Mathematics 25 (1979) 189-193. @I North-Holland Publishing Company NOTE MINIMUM BROADCAST GRAIPHS Arthur FARLEY, Stephen HEDETW ~%lI, Sandra MITCHELL and Andmej PROSKUROWSKI zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG University of Oregon, Eugene, OR 97403, U.S.A. Received 1 February 1978 Revised 10 August 1978 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1. Introduction Let a graph G = (V, E) represent a communication network. The set zyxwvutsrqponmlkjihg V of vertices corresponds to the members of the network, and the set E of edges corresponds to the communication lines connecting pairs of members. Suppose that a member originates a message which is to be communicated to all other members of the network. This is to be accomplished as quickly as possible by a series of calls placed over lines of the network. We adopt the constraints that (i) each call requires one unit of time, (ii) a member can participate in only one call per unit of time, and (iii) a member can only call an adjacent member. We refer to this one-to-all communication process as broadcasting. Consider the following problem: given a connected graph G and a meSSage originator, vertex u, what is the minimum number of time units required to complete broadcasting from vertex u? We define the broadcast time of u vertex u, b(u), to equal this minimum time. It is easy to see that for any vertex u in a connected graph G with n vertices, b(u) a [ log, nl , since during each time unit the number of informed vertices can at most double. We define the broadcast time of u graph G, 6(G), to equal the maximum broadcast time of any vertex u in G, i.e., b(G) = max {6(u) 1u E V(G)}. For the complete graph K, with n > 2 vertices, b(K,) = [ log, nl , yet K, is not minimal with respect to this property. Th’at is, we can remove several edges from & and still have a granh G such that b(G) = [ log, nl . We define a minimal broadcast graph to be a graph G with n vertices such that b(G) = [log, nl but for every proper spanning subgraph G’c G, b( G’) > [log* zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 4 l We define the broadcast function B(n) to equal the minimum number of edges in any minimal broadcast graph on n vertices. A minimum broadcast graph is a minimal broadcast graph on n vertices having B(n) edges. Frolm the point of view of applications, minimum broadcast graphs represent the cheapest possible cam- munication networks (in terms of number of lines) in which broadcasting can be accomplished from any vertex as fast as theoretically possible. 189