Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Committee Member

Dr. Kevin James, Committee Chair

Committee Member

Dr. Doug Rall

Committee Member

Dr. Wayne Goddard

Committee Member

Dr. Hui Xue


The 'domination chain,'' first proved by Cockayne, Hedetniemi, and Miller in 1978, has been the focus of much research. In this work, we continue this study by considering unique realizations of its parameters. We first consider unique minimum dominating sets in Cartesian product graphs. Our attention then turns to unique minimum independent dominating sets in trees, and in some direct product graphs. Next, we consider an extremal graph theory problem and determine the maximum number of edges in a graph having a unique minimum independent dominating set or a unique minimum maximal irredundant set of cardinality two. Finally, we consider a variation of domination, called identifying codes, in the Cartesian product of a complete graph and a path.