Threeway Submodularity in Connectivity Functions

Connectivity is a structural property of certain combinatorial objects in-cluding matroids and graphs. Connectivity functions provide a settingto prove and generalise statements regarding connectivity. A connectivityfunction is a normalized, symmetric and submodular integral set function.

We investigate the connectivity functions that satisfy a strictly strongercondition than submodularity called threeway submodularity. We deter-mine to the best of our abilities which connectivity functions satisfy thisproperty, and discuss the problem of recognizing it. We then provide amodular cut style theory for enumerating a class of connectivity functions.