On a Simple Depth-First Search Strategy for Exploring Unknown Graphs

We present a simple depth-rst search strategy for exploring (constructing) an unknown strongly connected graph G with m edges and n vertices by traversing at most min(mn; dn 2 + m) edges. Here, d is the minimum number of edges needed to add to G to make it Eulerian. This parameter d is known as the deeciency of a graph and was introduced by Kutten Kut88]. It was conjectured that graphs with low deeciency can be traversed more eeciently than graphs with high deeciency. Deng and Papadimitriou DP90] provided evidence that the conjecture may be true by exhibiting a family of graphs where the robot can be forced to traverse ? d 2 m edges in the worst case. Since then, there has been some interest in determining whether a graph with deeciency d can be explored by traversing O(poly(d)m) edges. Our algorithm achieves such bound when the graph is dense, say m = (n 2).