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A directedpseudoforestis a directed graph in which each vertex has at most one outgoing edge; that is, it has outdegree at most one.
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Apseudoforestis an undirected graph in which each connected component contains at most one cycle.
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Alternatively, omitting the self-loops produces a non-maximalpseudoforest.
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Almost every sufficiently sparse random graph ispseudoforest.
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Thus, a graph is apseudoforestif and only if it does not have the butterfly or the diamond as a minor.
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A pseudotree is a connectedpseudoforest.
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