  Graph cut problems are fundamental in Combinatorial Optimization, and are a
central object of study in both theory and practice. Furthermore, the study of
\emph{fairness} in Algorithmic Design and Machine Learning has recently
received significant attention, with many different notions proposed and
analyzed for a variety of contexts. In this paper we initiate the study of
fairness for graph cut problems by giving the first fair definitions for them,
and subsequently we demonstrate appropriate algorithmic techniques that yield a
rigorous theoretical analysis. Specifically, we incorporate two different
notions of fairness, namely \emph{demographic} and \emph{probabilistic
individual} fairness, in a particular cut problem that models disaster
containment scenarios. Our results include a variety of approximation
algorithms with provable theoretical guarantees.
