M. Studeny, T. Kroupa: Core-based criterion for extreme supermodular functions. Discrete Applied Mathematics 206 (2016), pp. 122-151.

Abstract
We give a necessary and sufficient condition for extremality of a supermodular function based on its min-representation by means of (vertices of) the corresponding core polytope. The condition leads to solving a certain simple linear equation system determined by the combinatorial core structure. This result allows us to characterize indecomposability in the class of generalized permutohedra. We provide an in-depth comparison between our result and the description of extremality in the supermodular/submodular cone achieved by other researchers.

AMS classification 68R05 91A12 90C27 68T30 52B12 52B40

Keywords
supermodular function
submodular function
core
conditional independence
generalized permutohedron
indecomposable polytope

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