difference between "minimal" and "minimum" edge cuts. I was going through the topic about connectivity of graphs. There it was mentioned about the terms "minimum edge cut" and "minimal edge cut". I know both are the sets of edges if removed from the graph $G$, makes $G$ disconnected. But I am unable to catch the basic difference betwen these two terms. Is minimal always minimum or vice versa? thanks.
 A: Usually the distinction is that a minimal example would be one that cannot be made smaller by taking a subset of the example's cuts, while a minimum is one that is as small as possible in absolute size.
For example, if we are talking about non-empty subsets of $\{1,2,\dots,n\}$ that add up to an even number, then $\{2\}$ is a minimum example and a minimal example, but $\{1,3\}$ is minimal but not a minimum.  There are examples which are smaller than $\{1,3\}$, but those examples are not contained in $\{1,3\}$.
A: See, for example, this link, which concisely lists the definitions and the distinction, and where you'll find illustrations depicting the distinctions.

An edge cut is a set of edges that, if removed from a connected graph,
  will disconnect the graph.
A minimal edge cut is an edge cut such that if any edge is put back in
  the graph, the graph will be reconnected.
A minimum edge cut is an edge cut such that there is no other edge cut
  containing fewer edges.  
A minimum edge cut is always minimal, but a minimal edge cut is not
  always minimum [bold face mine].
A minimal (and therefore minimum) edge cut will always yield two
  connected components.

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 $\qquad\qquad$ 
Figure $1$ shows the original graph.
Figure $2$ shows the maximum edge cut – just remove all edges.
Figure $3$ shows a minimum (and therefore minimal) edge cut.
Figure $4$ shows a minimal edge cut (which is not minimum).
