# 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 ?

• 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.