  An integral convex polytope ${\mathcal P}$ is said to be Gorenstein if its
toric ring $K[{\mathcal P}]$ is normal and Gorenstein. In this paper,
Gorenstein cut polytopes of graphs are characterized explicitly. First, we
prove that Gorenstein cut polytopes are compressed (i.e., all of whose reverse
lexicographic triangulations are unimodular). Second, by applying
Athanasiadis's theory for Gorenstein compressed polytopes, we show that a cut
polytope of a graph $G$ is Gorenstein if and only if $G$ has no $K_5$-minor and
$G$ is either a bipartite graph without induced cycles of length $\geq 6$ or a
bridgeless chordal graph.
