Difference Between Minimal and Minimum Edge Cuts

Question

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.

Answer 1

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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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).