In this paper we introduce a new notion of traversal sequences that we call exploration sequences. Exploration sequences share many properties with the traversal sequences defined in Aleliunas et al. (Proceedings on the 20th Annual Symposium of Foundations of Computer Science, 1979, pp. 218–223), but they also exhibit some new properties. In particular, they have an ability to backtrack, and their random properties are robust under choice of the probability distribution on labels.
Further, we present simple constructions of polynomial-length universal exploration sequences for some previously studied classes of graphs (e.g., 2-regular graphs, cliques, expanders), and we also present universal exploration sequences for trees. These constructions do not obey previously known lower bounds on the length of universal traversal sequences; thus, they highlight another difference between exploration and traversal sequences.
