A set of vertices D is a dominating set for a graph if every vertex is either in D or adjacent to a vertex which is in D. We show that the problem of finding a minimum dominating set in a chordal graph is NP-complete, even when restricted to undirected path graphs, but exhibit a linear time greedy algorithm for the problem further restricted to directed path graphs. Streamlined to handle only trees, our algorithm becomes the algorithm of Cockayne, Goodman and Hedetniemi. An interesting parallel with graph isomorphism is pointed out.