Enumeration and maximum number of minimal dominating sets for chordal graphs

Abstract We study the enumeration of the minimal dominating sets and upper bounds for the number of such sets in chordal graphs. We show that the maximum number of minimal dominating sets of an n -vertex chordal graph is at most 1.5048 n and prove that these sets can be enumerated in time O ( 1.5048 n ) . In this way we improve the previous upper bound of 1.5214 n , recently established by Abu-Khzam and Heggernes, and narrow the gap between the upper bound and the known lower bound of 1.4422 n .