Reachability Is Harder for Directed than for Undirected Finite Graphs (Preliminary Version)

Although it is known that reachability in undirected finite graphs can be expressed by an existential monadic second-order sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain “built-in” relations, such as the successor relation). The proof makes use of Ehrenfeucht-Frai’sse games, along with probabilistic arguments. However, we show that for directed finite graphs with degree at most k , reachability is expressible by an existential monadic second-order sentence. $