# Which cut does the “minimum cut” refer to?

My course notes give the following definitions; could someone please verify that the last definition is non-standard? (I've spent all evening googling, and isn't "minimum cut" a concept related to cut sets rather than to edge cuts (all terms as defined below)? For example, see http://en.wikipedia.org/wiki/Minimum_cut and http://scientopia.org/blogs/goodmath/2007/08/07/maximum-flow-and-minimum-cut/.)

A cut set of a graph $G$ induced by a partition of $G$'s vertices into sets $X$ and $Y$ is the set of all edges with one endpoint in $X$ and another endpoint in $Y$.

An edge cut of a connected graph $G$ is a set $S$ of $G$'s edges such that $G$-$S$ is disconected and $G$-$S$' is connected for any proper subset $S$' of $S$.

A minimum cut of a graph $G$ is an edge cut of $G$ with the smallest-possible cardinality (called the edge connectivity of $G$).

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Answering my own question: While cut set and edge cut do not mean the same thing, for a connected graph, the set of smallest possible cut sets and the set of smallest possible edge cuts actually do coincide, so both can be defined the same way and given the same name. While the 3rd definition above is not wrong, it is somewhat nonstandard, since "mimimum cut" is typically used in the context of network flows (eg. the max-flow-min-cut algorithm), where cut sets (which are not necessarily edge cuts) are a point of interest. – Ryan May 8 '13 at 22:29