How to correctly describe this action, deleting an edge that "shortcut" some vertices Haven't written a proof in years, not sure how to describe an algorithm like this ? Let us what we have a graph. like this below:



1). How to describe edge removal of{ (0, 1),(3,4), (1,2) }done in linear time ?  Basically, we don't want any edges that shortcuts our visit as every vertices need to be visited. 

Here is something:
Such edge removal can be achieved in linear time by scanning trough all the vertices in graph in the topological order, and for every outgoing edge(a, c) of v checking if the b comes after the vertex。

How can I correctly describe that if an algorithm remove (0,1),(3,4),(1,2) edges in this graph?
Your operation replaces your graph with one of its directed spanning trees. See for example lectuire notes of Uri Zwick.