Question

I am trying to find the maximal set for an undirected graph and here is the algorithm that i am using to do so:

1) Select the node with minimum number of edges
2) Eliminate all it’s neighbors
3) From the rest of the nodes, select the node with minimum number of edges
4) Repeat the steps until the whole graph is covered

Can someone tell me if this is right? If not, then why is this method wrong to calculate the maximal independent set in a graph?

Answer 1

What you have described will pick a maximal independent set. We can see this as follows:

1. This produces an independent set. By contradiction, suppose that it didn’t. Then there would have to be two nodes connected by edges that were added into the set you produced. Take whichever one of them was picked first (call it u, let the other be v) Then when it was added to the set, you would have removed all of its neighboring nodes from the set, including node v. Then v wouldn’t have been added to the set, giving a contradiction.

2. This produces a maximal independent set. By contradiction, suppose that it didn’t. This means that there is some node v that can be added to the independent set produced by your algorithm, but was not added. Since this node wasn’t added, it must have been removed from the graph by the algorithm. This means that it must have been adjacent to some node added to the set already. But this is impossible, because it would mean that the node v cannot be added to the produced independent set without making the result not an independent set. We have a contradiction.

Hope this helps!