What is a min-max theorem in graph theory? I'm currently studying a paper which uses extensively the term 'min-max theorems' in graph theory, and claims to present a tool allowing to generalize these theorems. (here is the link to the paper if needed)

Among those, we can find for example :


The max-flow-min cut theorem.
Edmond's disjoint arborescences theorem (link).


I have some intuition about what a min-max theorem would be, but I can't come with a concise and precise definition.

My question is : what would be a definition of such a family of theorems ?

And a second question along : is this min-max theorem concept always linked to the strong duality theorem, meaning that they mainly state that one problem is actually the dual of the other, like the max-flow-min-cut is ?
A min-max theorem is simply a theorem that says that the minimum value possible for one quantity is the maximum value possible for some other.
For example,


Max-flow min-cut says that the value of the biggest flow between two vertices in a weighted graph is equal to the value of the minimum cut that separates them.  Closely related, Menger's Theorem says that the maximum number of edge-disjoint paths between two vertex sets is equal to the size of the minimum cut that separates them.
Edmonds' Theorem says that the maximum number of edge-disjoint spanning arborescences is equal to the minimum value of something called $\varrho(X)$ over certain non-empty vertex sets.
The cops and robbers characterization of treewidth says that the minimum width over all tree decompositions is equal to the maximum number of cops that a robber can escape (er, plus or minus one, probably).