Minimum path cover--- Disjointed paths with minimum total number of edges
A Trivial Algorithm

You can compute the lengths of all-pair shortest paths among $\mathcal{V}'$, and solve a minimum perfect matching in a complete graph on $\mathcal{V}'$ where the weight of an edge $(u,v)$ is the length of the shortest path from $u$ to $v$.

A Smarter Algorithm

Consider the connected component containing $\mathcal{V}'$, it must be a tree.

At each step we move the deepest nodes in $\mathcal{V}'$ to their fathers. If two nodes move to the same node, we decide to match them and do not consider them any more in following steps. If more than two nodes move to the same node, repeatedly match arbitrary two of them until no more than one node is left.

This algorithm runs in $O(|\mathcal{E}|)$.