Graphs with Unique Minimum Specified Domination Sets

In this paper, we consider the graphs that have a unique set that achieves a domination parameter and determine the maximum value of the associated parameter, both in graphs in general and in graphs without end-vertices. For example, for a graph G of order n without an end-vertex, we show an upper bound of n/3 if G has a unique minimum dominating set, an upper bound of (n- 1 ) / 2 if G has a unique minimum total dominating set or a unique minimum 2-dominating set, and an upper bound of n/2 if G has a unique paired dominating set.