A 1-relaxed minimum broadcast graph on 15 vertices Original Research Article

In 1998, Shastri studied the sparsest possible broadcast graphs in which broadcasting can be accomplished in slightly more than the optimal time of View the MathML source. In particular, they constructed the sparsest possible time-relaxed broadcast graphs for small n (≤ 14) and very sparse time-relaxed broadcast graphs for larger n (≤ 65). Let Bt(n) be the number of edges in the sparsest possible graph on n vertices in which broadcasting can be accomplished in t additional steps than the optimal (i.e., in View the MathML source steps), they conjectured that B1(15) = 18. In this paper, we give a 1-relaxed minimum broadcast graph on 15 vertices which shows that B1(15) = 17, thus reject the conjecture.