We describe the class of quiver settings with one dimensional vertices whose semi-simple representations are parametrized by a complete intersection variety. We show that these quivers can be reduced to a one vertex quiver with some combinatorial reduction steps. We introduce the notion of descendants which is similar to graph theoretic minors, and show that the class in question can be characterized as the quivers that do not have two specific forbidden descendants. We also prove that the class of coregular quiver settings with arbitrary dimension vector, which has been understood earlier via reduction steps, can also be described as the ones that do not contain a certain forbidden descendant.
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