A *digraph* (direct graph) consists of a set $V$ of vertices and a set $E$ of directed edges $v\to v'$. A *multidigraph* is a digraph in which $E$ is a multiset, so edges may appear multiple times in $E$, or equivalently, $E$ is a set of directed edges that are assigned multiplicities.

Is there a standard name for a multidigraph with the following property? For each vertex $v\in V$, there is at most one $v'\in V$ such that the edge $v\to v'$ is in $E$, although that edge is of course allowed to appear multiple times in $E$.

I looked through the list of various types of graphs on Wikipedia, but didn't see this one. (For a digraph that's not "multi", one could say "digraph with all out-degrees 0 or 1," but even that seems clumsy.)

linear digraphis a digraph in which each vertex has at most one out-arrow. (Also, just for curiosity, why do you say "a collection of"? Does the word "graph" imply connected?) $\endgroup$ – Joe Silverman Jun 13 '18 at 15:36