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Is there any special word for a maximal non-branching directed path in a network or diforest?

To be 100% precise, by "maximal non-branching directed path" I mean a path $P=x_1,x_2,\ldots$ (maybe finite, maybe infinite) such that

  1. Every vertex in the path (except possibly the terminal vertex of the path (if there is one)) has outdegree 1.
  2. The path is directed (for every $x_i,x_{i+1}$ in the path, there is an edge directed from $x_i$ to $X_{i+1}$).
  3. There is no $x_0$ such that $P'=x_0,x_1,\ldots$ has properties 1--2.
  4. If $P$ is finite (say $P=x_1,\ldots,x_n$) then there is no $x_{n+1}$ such that $x_1,\ldots,x_{n+1}$ has properties 1--2.

If I were king of the world I would decree that such things are "segments", but that terminology doesn't seem to exist in the literature.

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