A new construction of broadcast graphs

Abstract

Given a graph G=(V,E) and an originator vertex v, broadcasting is an information disseminating process of transmitting a message from the vertex v to all vertices of the graph G as quickly as possible. A graph G on n vertices is called broadcast graph if the broadcasting from any vertex in the graph can be accomplished in ⌈logn⌉ time. A broadcast graph with the minimum number of edges is called minimum broadcast graph. The number of edges in a minimum broadcast graph on n vertices is denoted by B(n). A long sequence of papers present different broadcast graph constructions and upper bounds on B(n). In this paper, we improve the compounding method and construct new broadcast graphs with a better upper bound on B(n). Consequently, we show that B(n)≤(m−k+1)n−(2m−k+1−2)(m−2q+1)2q−1 for n∈[2m−1+1,2m−2m2+1], where n=2m−2k−d, m≥5, 2≤k≤m−2, 0≤d≤2k−1, and q=min(⌊m−22⌋,k−2).