Exploring an unknown graph

It is desired to explore all edges of an unknown directed, strongly connected graph. At each point one has a map of all nodes and edges visited, one can recognize these nodes and edges upon seeing them again, and it is known how many unexplored edges emanate from each node visited. The goal is to minimize the ratio of the total number of edges traversed to the optimum number of traversals had the graph been known. For Eulerian graphs this ratio cannot be better than 2, and 2 is achievable by a simple algorithm. In contrast, the ratio is unbounded when the deficiency of the graph (the number of edges that have to be added to make it Eulerian) is unbounded. The main result is an algorithm that achieves a bounded ratio when the deficiency is bounded; unfortunately the ratio is exponential in the deficiency. It is also shown that, when partial information about the graph is available, minimizing the worst-case ratio is PSPACE-complete