How to find the minimum connectors to connect selected vertices (subset of vertices)

Question

suppose that i have a un-directed graph with 10 vertices and 24 edges which connects with each other, as an example if i give an input like this (this is not the actual one, actual one is 10*10)

int graph[][] = new int[][]{{0, 0, 0, 0, 0, 0},
                            {0, 0, 2, 0, 6, 0},
                            {0, 2, 0, 3, 8, 5},
                            {0, 0, 3, 0, 0, 7},
                            {0, 6, 8, 0, 0, 9},
                            {0, 0, 5, 7, 9, 0}};

so i want to find the minimum connectors in order to connect specific vertices ex: vertices 0,2,4 . currently i applied the prims algorithm in order to get the minimum connectors through out all the edges of the graph, but as i don't want all the edges to be included in my MST i need some help achieving this.

• I can't remove the edges what i don't want in the beginning because it will disconnect the graph

please help.

Answer 1

So you have 6 vertices and 7 undirected weighted edges:

 1 ←→ 2 (w=2)
 1 ←→ 4 (w=6)
 2 ←→ 3 (w=3)
 2 ←→ 4 (w=8)
 2 ←→ 5 (w=5)
 3 ←→ 5 (w=7)
 4 ←→ 5 (w=9)

You need to connect vertices 0, 2, and 4, so you build a new graph with those 3 vertices, and find the shortest paths between them.

 0 ←→ 2 (no edge)
 0 ←→ 4 (no edge)
 2 ←→ 4 (w=8)

That of course cannot be done, since there are no edges to vertex 0.

So let's try vertices 1, 3, and 5 instead:

 1 ←→ 3 (w=5, 1 ←→ 2 ←→ 3)
 1 ←→ 5 (w=7, 1 ←→ 2 ←→ 5)
 3 ←→ 5 (w=7, 3 ←→ 5)

Now apply a minimum spanning tree (MST) to that:

 1 ←→ 3 ←→ 5 (w=12)

Expanding back to original graph:

 1 ←→ 2 ←→ 3 ←→ 5 (w=12)