  We consider the problem of exploration of networks, some of whose edges are
faulty. A mobile agent, situated at a starting node and unaware of which edges
are faulty, has to explore the connected fault-free component of this node by
visiting all of its nodes. The cost of the exploration is the number of edge
traversals. For a given network and given starting node, the overhead of an
exploration algorithm is the worst-case ratio (taken over all fault
configurations) of its cost to the cost of an optimal algorithm which knows
where faults are situated. An exploration algorithm, for a given network and
given starting node, is called perfectly competitive if its overhead is the
smallest among all exploration algorithms not knowing the location of faults.
We design a perfectly competitive exploration algorithm for any ring, and show
that, for networks modeled by hamiltonian graphs, the overhead of any DFS
exploration is at most 10/9 times larger than that of a perfectly competitive
algorithm. Moreover, for hamiltonian graphs of size at least 24, this overhead
is less than 6% larger than that of a perfectly competitive algorithm.
