## Reachability is harder for directed than for undirected finite graphs (1990)

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Venue: | Journal of Symbolic Logic |

Citations: | 71 - 8 self |

### BibTeX

@ARTICLE{Ajtai90reachabilityis,

author = {Miklos Ajtai and Ronald Fagin},

title = {Reachability is harder for directed than for undirected finite graphs},

journal = {Journal of Symbolic Logic},

year = {1990},

volume = {55},

pages = {113--150}

}

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### Abstract

Abstract. 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. $1. Introduction. If s and t denote distinguished points in a directed (resp. undirected) graph, then we say that a graph is (s, t)-connected if there is a directed (undirected) path from s to t. We sometimes refer to the problem of deciding whether a given directed (undirected) graph with two given points sand t is (s, t)-connected as the directed (undirected) reachability problem.