On Kernel Mengerian Orientations of Line Multigraphs

TL;DR

This paper characterizes kernel properties in orientations of line multigraphs, establishing equivalences among kernel perfect, kernel ideal, and kernel Mengerian orientations, thus generalizing previous theorems in graph theory.

Contribution

It provides a polyhedral description of kernels in line multigraph orientations and proves the equivalence of several kernel-related properties, extending known results.

Findings

Kernel perfect, kernel ideal, and kernel Mengerian orientations are equivalent in line multigraphs.

The result generalizes the theorem of Borodin et al. on kernel perfect digraphs.

It extends the theorem of Kiraly and Pap on stable matching problem.

Abstract

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.