Constructing Independent Spanning Trees on Bubble-Sort Networks

Abstract

A set of spanning trees in a graph G is called independent spanning trees (ISTs for short) if they are rooted at the same vertex, say r, and for each vertex \(v(\ne r)\) in G, the two paths from v to r in any two trees share no common vertex except for v and r. Constructing ISTs has applications on fault-tolerant broadcasting and secure message distribution in reliable communication networks. Since Cayley graphs have been used extensively to design interconnection networks, the study of constructing ISTs on Cayley graphs is very significative. It is well-known that star networks \(S_n\) and bubble-sort network \(B_n\) are two of the most attractive subclasses of Cayley graphs. Although it has been dealt with about two decades for the construction of ISTs on \(S_n\) (which has been pointed out that there is a flaw and has been corrected recently), so far the problem of constructing ISTs on \(B_n\) has not been dealt with. In this paper, we present an efficient algorithm to construct \(n-1\) ISTs of \(B_n\). It seems that our work is the latest breakthrough on the problem of ISTs for all subclasses of Cayley graphs except star networks.