Oblivious periodic traversal of anonymous, undirected graphs with advice and adversarial port numbers

We consider the problem of periodic graph traversal [2], which has previously been studied in a variety of settings [2, 5, 3, 7, 13, 12, 10, 11]. The problem of periodic graph traversal is concerned with an agent having to visit every node of a graph and return to its start location and state. The periodic graph traversal problem can be extended by labeling each node to help the agent explore the graph [3]. We consider oblivious agents (agents with no persistent memory) and graphs with low maximum degree. Ports (the points where edges are incident to vertices) are numbered by an adversary. We prove that two labels are both necessary and su cient to explore all degree three bounded graphs and prove that there are only two (trivially isomorphic) agent algorithms that can explore all such graphs. We provide a two label labeling algorithm that allows these agent algorithms to explore all such graphs in a period of length 4n−2 where n is the number of vertices in the graph. For the special case of Hamiltonian graphs, which includes almost all regular graphs [17], we provide an algorithm that improves the period length to n using ∆(G) labels. In the case of degree three bounded graphs, we provide a 5-label algorithm that improves the period length to 2 12n− 1. In the general case of all graphs, the best possible worst-case period length is 2n−2. We give a 2−1 label algorithm (7-label for degree three bounded graphs) that achieves this for all graphs. Finally we extend our results to graphs of higher degree and prove upper bounds of 6 and 24 on the number of labels required for an agent ∗Supported in part by NSF grant CCF-0916389