On Kernel Mengerian Orientations of Line Multigraphs

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.