SAMS Subject Classification Number: 6
We introduce an intersection graph of a connected graph \(G\), the mincut graph of G, such that every minimum edge-cut in \(G\) is a vertex in the mincut graph, \(X(G\)), and two vertices in \(X(G)\) are adjacent if their corresponding edge-cuts in G share at least one edge. We present the mincut graph as a graph operator. We ask and attempt to answer questions such as ‘Which graphs appear as images of graphs?’; ‘Which graphs are fixed under the operator?’; ‘What happens if the operator is iterated?’ Thus, we show that every graph is a mincut graph and furthermore, that non-isomorphic graphs can have isomorphic mincut graphs. We also characterise graphs fixed under the operator and show that no graph diverges under iteration of the operator.