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I am a software engineer. I'm dealing with a data structure which represents a digraph of a very specific structure. I am wondering if this class of graph has been identified and studied as I need to do a fair bit of work with it and would love to not reinvent the wheel if not necessary.

The structure of the graph is as follows. It is a digraph wherein each vertex has at most one edge leading from it. Note that a vertex is allowed to have zero edges leading from it as well. There is no restriction on the edges that lead into a vertex, nor is there a restriction on the graph being cyclic.

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If you needed to compute iterates, find connected components, or store it efficiently, you might mention that. Otherwise, it is not clear how an answer will help, as most optimized algorithms handle more general cases depending on the action desired. Gerhard "Email Me About System Design" Paseman, 2011.07.14 –  Gerhard Paseman Jul 14 '11 at 19:02

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up vote 7 down vote accepted

Qiaochu Yuan has already provided an answer in terms of functions, but if you prefer to think graph-theoretically these things are known as directed pseudoforests.

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If I understand you correctly, such a thing is generally called a partial function (from the vertex set to itself).

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