Linear Algorithm for Broadcasting in Unicyclic Graphs

Abstract

Broadcasting is an information dissemination problem in a connected network, in which one node, called the originator, disseminates a message to all other nodes by placing a series of calls along the communication lines of the network. Once informed, the nodes aid the originator in distributing the message. Finding the minimum broadcast time of a vertex in an arbitrary graph is NP-complete. The problem is solved polynomially only for trees. It is proved that the complexity of the problem of determining the minimum broadcast time of any vertex in an arbitrary tree T = (V,E) is Θ|V|. In this paper we present an algorithm that determines the broadcast time of any originator in an arbitrary unicyclic graph G = (V,E) in O(|V|) time. This, combined with the obvious lower bound, gives a Θ(|V|) solution for the problem of broadcasting in unicyclic graphs. As a byproduct, we also find a broadcast center of the unicyclic graph (a vertex in G with the minimum broadcast time).