Minimum damaging costs in graph

We are given a graph G(V,E) with N nodes (Numbered from 0 to N-1) and exactly (N-1) two-way Edges.

Each edge in a graph has a positive cost C(u,v)(Edge weight).

The entire graph is such that there is a unique path between any pair of Nodes.

We are also given a List L of node number,at which bomb are placed.

Our aim is to damage/remove the edge from the graph such ,that after damaging/removing the edges from the graph ,there is no connection among the Bombs --

that is after damaging, there is no path between any two bombs.

The cost of damaging the Edge(u,v) = Edge weight(u,v).

So, we have to damage those edges, such that the total damaging cost is minimum.

Example:

Total Nodes=N=5 
Total Bomb=Size of List L=3

List L={2,4,0}//Thats is at node number 2,4 and 0 bomb is placed...

Total Edges =N-1=4 edges are::

u v Edge-Weight

2 1 8

1 0 5

2 4 5

1 3 4



In this case the answer is ::
Total Damaging cost =(Edge Weight (2,4) + Edge Weight(0,1))
           =5+5=10.

So when we remove the edge connecting node (2,4),
and the edge connecting node (0,1) ,there is no connection left 
between any pair of machines in List {2,4,0};

Note any other,combinations of edges(that  we damaged ) to achieve the
target goal ,needs more than 10 unit cost.  

Constraints::
N(ie. Number of Nodes) <= 100,000
ListSize |L|(ie. Number of Bombs) <= N
1 <=Edge cost(u,v) <= 1000,000

What i had done?

Until now, I had not found any efficient way :( .

Further, as the number of nodes is N, the number of edges is exactly N-1 and the entire graph is such there is a Unique path between any pair of Nodes, I got a conclusion that the graph is a TREE.

I tried to modify the Kruskal algorithm but that didn't help me either.

Thanks!

This is the Multiway Cut problem in trees. It can be solved in polynomial time by a straightforward dynamic programming. See Chopra and Rao: "On the multiway cut polyhedron", Networks 21(1):51–89, 1991.