  We present a polyhedral description of kernels in orientations of line
multigraphs. Given a digraph $D$, let $FK(D)$ denote the fractional kernel
polytope defined on $D$, and let ${\sigma}(D)$ denote the linear system
defining $FK(D)$. A digraph $D$ is called kernel perfect if every induced
subdigraph $D^\prime$ has a kernel, called kernel ideal if $FK(D^\prime)$ is
integral for each induced subdigraph $D^\prime$, and called kernel Mengerian if
${\sigma} (D^\prime)$ is TDI for each induced subdigraph $D^\prime$. We show
that an orientation of a line multigraph is kernel perfect iff it is kernel
ideal iff it is kernel Mengerian. Our result strengthens the theorem of Borodin
et al. [3] on kernel perfect digraphs and generalizes the theorem of Kiraly and
Pap [7] on stable matching problem.
