On a directed graph G a depth first search $($DFS$)$ was run. It was started at node A$,$ and when there were more than one possibilities for the next vertex, alphabetically the first was chosen. Edges enter the DFS tree in the following order: AB$,$ BE$,$ EG$,$ EH$,$ AC$,$ CF$,$ AD$.$

How many edges among directed edges EA$,$ EC$,$ HF$,$ BG$,$ DG may be present in graph G$?$