Sink-Stable Sets of Digraphs

TL;DR

This paper introduces sink-stable sets in digraphs, establishes a min-max formula for their maximum union size, and connects these results to existing theorems in graph theory, including coloring and stable set theorems.

Contribution

It presents a new concept of sink-stable sets, proves a min-max theorem for their union, and links these findings to several important existing graph theory results.

Findings

Proved a min-max formula for sink-stable sets

Connected sink-stable sets to Clar number of bipartite plane graphs

Sharpened Minty's coloring theorem

Abstract

We introduce the notion of sink-stable sets of a digraph and prove a min-max formula for the maximum cardinality of the union of k sink-stable sets. The results imply a recent min-max theorem of Abeledo and Atkinson on the Clar number of bipartite plane graphs and a sharpening of Minty's coloring theorem. We also exhibit a link to min-max results of Bessy and Thomasse and of Sebo on cyclic stable sets.