Cut dominants and forbidden minors

TL;DR

This paper characterizes graphs whose cut dominant polyhedron can be described with inequalities having small integer coefficients, linking to TSP-perfect graphs and providing insights into forbidden minors for various right-hand sides.

Contribution

It provides a forbidden-minor characterization for graphs with cut dominants defined by inequalities with small integer coefficients, extending understanding of TSP-perfect graphs.

Findings

Characterization of graphs with cut dominant inequalities with RHS ≤ 2

Shorter proof of the relation to TSP-perfect graphs

Properties of forbidden minors for RHS > 2

Abstract

The cut dominant of a graph is the unbounded polyhedron whose points are all those that dominate some convex combination of proper cuts. Minimizing a nonnegative linear function over the cut dominant is equivalent to finding a minimum weight cut in the graph. We give a forbidden-minor characterization of the graphs whose cut dominant can be defined by inequalities with integer coefficients and right-hand side at most 2. Our result is related to the forbidden-minor characterization of TSP-perfect graphs by Fonlupt and Naddef (Math. Prog., 1992). We prove that our result implies theirs, with a shorter proof. Furthermore, we establish general properties of forbidden minors for right-hand sides larger than 2.