I am trying to figure out a greedy algorithm that finds the optimum (minimum) dominating set for any tree in linear time.
So a greedy algorithm to find a dominating set for a general graph is not optimum. It's an approximation of the optimum dominating set. But since this is a tree I am assuming that a greedy algorithm can give the optimum.
What i have so far is:
Select a vertex with the maximum number of adjacent vertices that are not dominated (that is, it's neighbors are either not dominated by one it its neighbors or they are not a dominated vertex themselves). We add this vertex to the dominated set.
We repeat this proceudue until all vertices are either in the dominated set or are neighbors of one of the dominated vertices.
But I am not sure this will give an optimum solution.