Submodular functions have appeared to be a key tool for
proving the monotonicity of several graph searching games.
In this paper we provide a general game theoretic framework able to
unify old and new monotonicity results in a unique min-max
theorem. Our theorem, provides a game theoretic analogue
to a wide number of graph theoretic parameters such as
linear-width and cutwidth.
Fomin, F., Thilikos, D. "On the monotonicity of games generated by symmetric submodular functions". 2001.