An algebraic analysis of the connectivity of DeBruijn and shuffle-exchange digraphs

Abstract

We study connectivity properties of d-ary deBruijn and shuffle-exchange digraphs by appealing to their algebraic structure. Our first result proves that both these families of digraphs are (d − 1)-connected. The proof also leads to two substantially stronger results. Namely, we prove that for the order-n, d-ary de Bruijn digraph (resp. the order-n, d-ary shuffle-exchange digraph), any set of shuffle cyclesof total length less than n(d − 1) can be removed and the digraph remains strongly connected. The second extension characterizes the pairs of vertices in the d-ary deBruijn digraphs (resp. the d-ary shuffle-exchange digraphs) which have d disjoint paths between them. The central idea in the paper rests upon a new application of the group-theoretic relationship between shuffle-oriented digraphs, butterfly-like digraphs and hypercubes.