  We study a randomized algorithm for graph domination, by which, according to
a uniformly chosen permutation, vertices are revealed and added to the
dominating set if not already dominated. We determine the expected size of the
dominating set produced by the algorithm for the path graph $P_n$ and use this
to derive the expected size for some related families of graphs. We then
provide a much-refined analysis of the worst and best cases of this algorithm
on $P_n$ and enumerate the permutations for which the algorithm has the
worst-possible performance and best-possible performance. The case of
dominating the path graph has connections to previous work of Bouwer and Star,
and of Gessel on greedily coloring the path.
