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Reachability is harder for directed than for undirected finite graphs. (English) Zbl 0708.03016
A directed (undirected) graph is (s,t)-connected if there is a directed (undirected) path from s to t. Refer to the problem of deciding whether a given directed (undirected) graph with two given points s and t is (s,t)- connected as the directed (undirected) reachability problem. P. Kanellakis observed that undirected reachability is monadic \(\Sigma^ 1_ 1\) and posed as an open problem the question of whether directed reachability is monadic \(\Sigma^ 1_ 1\). The main result in this paper is that directed reachability is not monadic \(\Sigma^ 1_ 1\) (even in the presence of certain “built-in” relations, such as the successor relation). The proof makes use of Ehrenfeucht-Fraïssé games, along with probabilistic arguments. In addition, this paper proves that for directed finite graphs with degree at most k, reachability is monadic \(\Sigma^ 1_ 1\).
Reviewer: Tao Renji

MSC:
03D15 Complexity of computation (including implicit computational complexity)
05C40 Connectivity
68Q25 Analysis of algorithms and problem complexity
68R10 Graph theory (including graph drawing) in computer science
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