Constructing Independent Spanning Trees in Alternating Group Networks

Abstract

In a graph G, two spanning trees \(T_1\) and \(T_2\) are rooted at the same vertex r. If for every \(v \in V(G)\) the paths from v to the root r in \(T_1\) and \(T_2\) are internally vertex-disjoint, they are independent spanning trees (ISTs). ISTs have numerous applications, such as secure message distribution and fault-tolerant broadcasting. The alternating group network \(AN_n\) (n stands for the dimension) is a subclass of Cayley graphs, and the approach of constructing ISTs in \(AN_n\) has not been proposed until now. In this paper, we propose a recursive algorithm for constructing ISTs in \(AN_n\). The algorithm is a top-down approach, and the parent of one node in an IST is not determined by any rule. The correctness of the algorithm is verified, and the time complexity is analyzed. We use PHP to implement the algorithm and test cases from \(AN_3\) to \(AN_{10}\). The testing results show that all trees are ISTs in all cases. We conclude that our algorithm is not only correct but also efficient.