  A $k$-submodular function is a generalization of submodular and bisubmodular
functions. This paper establishes a compact representation for minimizers of a
$k$-submodular function by a poset with inconsistent pairs (PIP). This is a
generalization of Ando-Fujishige's signed poset representation for minimizers
of a bisubmodular function. We completely characterize the class of PIPs
(elementary PIPs) arising from $k$-submodular functions. We give algorithms to
construct the elementary PIP of minimizers of a $k$-submodular function $f$ for
three cases: (i) a minimizing oracle of $f$ is available, (ii) $f$ is
network-representable, and (iii) $f$ arises from a Potts energy function.
Furthermore, we provide an efficient enumeration algorithm for all maximal
minimizers of a Potts $k$-submodular function. Our results are applicable to
obtain all maximal persistent labelings in actual computer vision problems. We
present experimental results for real vision instances.
