The mincut graph of a graph

In this paper we introduce an intersection graph of a graph G , with vertex set the minimum edge-cuts of G . We find the minimum cut-set graphs of some well-known families of graphs and study the mincut graph as a graph operator. In doing so we follow the research programme on graph operators, as introduced by Prisner in the 1995 monograph "Graph Dynamics". Thus we ask and attempt to answer questions such as 'Which graphs appear as images of graphs?'; 'Which graphs are fixed under the operator?'