Is there a name for the class of directed acyclic graphs where, for every node with more than one predecessor, none of the direct predecessors are in any of the others' ancestry? I.e. the most recent common ancestor of any pair of a node's direct predecessors is not either of the pair.
I have found the concepts of mangroves and polytrees, but do not understand very well whether they are applicable here. I believe polytrees are a more restrictive concept than what I am looking for, because I specifically do want to allow a node to have multiple "siblings" (from a common grandparent) as direct predecessors, just so long as it does not directly link to the grandparent or any of its ancestors.
I'm specifically interested in the case where there is a root but I don't expect that to make much difference to the above.
As an example, using this picture from Wikipedia 
This graph does not meet my criteria because the D node connects to B (or equivalently, C) and also A, which is B's predecessor. The node E is also triply disqualifying because it links to D and C, D's predecessor, but also links to both C and A, C's predecessor. It would still be disqualifying if the link C->E was removed because A is a predecessor of D. This would meet my criteria if the link from D to A was removed, and any single one of the links to E was retained and the rest removed. Additionally (although I'm less sure about this) , if we retain only the link D->E, there is no way to add another single node to the graph with more than one parent/predecessor without breaking the criteria above, because everything would be an ancestor of E.