On a simple depth-first search strategy for exploring unknown graphs

Abstract

We preset a siple epth-first search strategy for exploring (constructing) an unknown strongly connected graph G with m edges and n vertices by traversing at most min (mn,dn2 + 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 deficiency of a graph and was introduced by Kutten [Kut88]. It was conjectured that graphs with high deficiency. 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 Ω (d2 m) edges in the worst case. Since then, there has been some interest in determining whether a graph with deficiency d can be explored by traversing O(poly(d)m) edges. Our algorithm achieves such bound when the graph is dense, say m = Ω(n2).