How to turn an undirected graph with multi source to an DAG?

Question

I have a undirected acyclic graph, which expressed by adjacency list, such as below.

{A: C}
{B: D}
{C: A,D}
{D: C,B,E}
{E: D,F,G}
{F: E}
{G: E}

My problem is: above undirected acyclic grapgh has multi independent source vertexes, which are A and B, and multi independent sink vertexes, which are F and G. How can I turn it to an DAG? The DAG can be expressed by adajacency list below

{A: C}
{B: D}
{C: D}
{D: E}
{E: F, G}

I have an simple solution with normal DFS below. However, is there any more efficiant solution or algorithm?

1. For each source vertex, do DFS and will find the only shortest path that lead to the sink vertices.
2. Add above path to the final DAG
3. Repeat steps 1 and 2 for each source vertex