We study a new version of the domination problem in which the dominating set is required to be a clique. The minimum dominating clique problem is NP-complete for split graphs and, hence, for chordal graphs. We show that for two other important subclasses of chordal graphs the problem is solvable efficiently. We present an O(m · logn) algorithm for strongly chordal graphs and an O(n4) algorithm for undirected path graphs.
