Universal traversal sequences for paths and cycles

Abstract

One approach to the reachability problem for rooted undirected graphs G is to order the neighbors of each vertex and traverse all vertices of the connected component containing the root by means of a sequence of positive integers ( s i ) i ≥ 1 interpreted as instructions of the form, “Move to the s i th neighbor or of the current vertex.” A sequence that does this for all d-regular connected graphs with n vertices is called an ( n, d)-universal traversal sequence. Although the existence of universal traversal sequences is easy to verify, known methods for their construction involve some sort of exhaustive search. In this paper, a recursive algorithm for the construction of ( n, 2)-universal traversal sequences in space O(log 2 n) is described.