Reachability is harder for directed than for undirected finite graphs

It is shown that for directed graphs, reachability can not be expressed by an existential monadic second-order sentence. The proof makes use of Ehrenfeucht-Fraisse games, along with probabilistic. However, it is shown that for directed graphs with degree at most k, reachability is expressible by an existential monadic second-order sentence. One reason for the interest in the main result is that while there is considerable empirical evidence (in terms of the efficiency of algorithms that have been discovered) that reachability in directed graphs is 'harder' than reachability in unidirected graphs, this is the first proof in a precise technical sense that this is so.