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There is a famous classification of the path algebras of finite acyclic quivers into finite, tame, and wild representation types. For quivers with cycles, it is standard that the 2-loop quiver (with only one vertex) is wild (perhaps the original wild problem?), and then it is not too hard to show that any quiver with two cycles is wild as well.

On the other hand, the one-loop quiver is easily seen to be tame (again, perhaps the canonical example of a tame classification problem is Jordan normal form), as are the cycle graphs on $n$ vertices (consisting of a single $n$-cycle, also known in this context as the cyclic orientation of the extended Dynkin quiver $\tilde A_n$).

But what about other finite unicyclic graphs (=graphs that contain a single cycle)?

(I spent quite some time searching for this, both online and in books, as well as a little time thinking about it, without success.)

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Dynkin quivers are of finite type, Euclidean quivers are of tame type, and all other quivers are wild. In particular, if there is a Euclidean proper subquiver then the representation type is wild. A cycle is an $\tilde A_n$ subquiver. So if a quiver contains a cycle but is not a cycle then it is wild.

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    $\begingroup$ See for example Proposition 4.7.1 in my Representations and Cohomology I. $\endgroup$
    – Dave Benson
    Commented May 24, 2023 at 22:34

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