  There is a well-documented research programme on graph operators which
addresses questions such as `Which graphs appear as images of graphs?'; `Which
graphs are fixed under the operator?'; `What happens if the operator is
iterated?' In this paper, we introduce an intersection graph called the mincut
graph and state its characteristics. We address some of the research programme
questions for the mincut graph operator and conclude by stating further
research questions on the mincut graph.
