A graph G is referred to as domination vertex critical if the removal of any vertex results in a reduction of the domination number. It is considered dot-critical (or totally dot-critical) if contracting any edge (or identifying any two vertices) leads to a decrease in the domination number. In this concise paper, we delve into the investigation of these properties and proceed to characterize the dot-critical and totally dot-critical attributes of Harary graphs.
