The minimum spanner problem on butterfly graphs

Given a connected graph G, a spanning subgraph G´ of G is called a t-spanner if every pair of two adjacent vertices in G has a distance of at most t in G´. A t-spanner of a graph G is minimum if it contains minimum number of edges among all t-spanners of G. Finding minimum spanners for general graphs is rather difficult. Most of the previous results were obtained for some particular graphs, e.g., butterfly graphs, cube-connected cycles, de Bruijin graphs, Kautz graphs, complete bipartite graphs and permutation graphs. The butterfly graphs were originally introduced as the underlying graphs of FFT networks which can perform the fast Fourier transform (FFT) very efficiently. We successfully construct most of the minimum t-spanners for the k-ary r-dimensional butterfly graphs for 2⩽t⩽6 and t=8