Characterizing N+-perfect line graphs

TL;DR

This paper investigates the properties of N+-perfect line graphs, confirming that their stable set polytope is characterized solely by cliques and odd holes, supporting a broader conjecture about N+-perfect graphs.

Contribution

It proves that in N+-perfect line graphs, the only facet-defining graphs are cliques and odd holes, advancing understanding of N+-perfection in specific graph classes.

Findings

N+-perfect line graphs have only cliques and odd holes as facet-defining graphs.

Supports the conjecture that N+-perfect graphs are characterized by near-bipartite support inequalities.

Verifies the conjecture for line graphs, a significant class in graph theory.

Abstract

The subject of this contribution is the study of the Lov\'asz-Schrijver PSD-operator N+ applied to the edge relaxation of the stable set polytope of a graph. We are particularly interested in the problem of characterizing graphs for which N+ generates the stable set polytope in one step, called N+-perfect graphs. It is conjectured that the only N+-perfect graphs are those whose stable set polytope is described by inequalities with near-bipartite support. So far, this conjecture has been proved for near-perfect graphs, fs-perfect graphs, and webs. Here, we verify it for line graphs, by proving that in an N+-perfect line graph the only facet-defining graphs are cliques and odd holes.