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 (si)i ≥ 1 interpreted as instructions of the form, “Move to the sith 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(log2n) is described.