The mincut graph of a graph (original) (raw)

In this paper we introduce an intersection graph of a graph GGG, with vertex set the minimum edge-cuts of GGG. We find the minimum cut-set graphs of some well-known families of graphs and show that every graph is a minimum cut-set graph, henceforth called a \emph{mincut graph}. Furthermore, we show that non-isomorphic graphs can have isomorphic mincut graphs and ask the question whether there are sufficient conditions for two graphs to have isomorphic mincut graphs. We introduce the rrr-intersection number of a graph GGG, the smallest number of elements we need in SSS in order to have a family F=S1,S2ldots,SiF=\{S_1, S_2 \ldots , S_i\}F=S1,S2ldots,Si of subsets, such that ∣Si∣=r|S_i|=r∣Si∣=r for each subset. Finally we investigate the effect of certain graph operations on the mincut graphs of some families of graphs.