On Box-Perfect Graphs

TL;DR

This paper proves the Cameron-Edmonds conjecture on box-perfect graphs, characterizes several classes of such graphs, and introduces a new method for establishing box-perfectness, advancing understanding in graph theory and combinatorial optimization.

Contribution

It proves the Cameron-Edmonds conjecture for parity graphs, characterizes multiple classes of box-perfect graphs, and develops a general method for proving box-perfectness.

Findings

Proof of Cameron-Edmonds conjecture for parity graphs

Identification of new classes of box-perfect graphs

Development of a general method for establishing box-perfectness

Abstract

Let $G=(V,E)$ be a graph and let $A_G$ be the clique-vertex incidence matrix of $G$. It is well known that $G$ is perfect iff the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is totally dual integral (TDI). In 1982, Cameron and Edmonds proposed to call $G$ box-perfect if the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is box-totally dual integral (box-TDI), and posed the problem of characterizing such graphs. In this paper we prove the Cameron-Edmonds conjecture on box-perfectness of parity graphs, and identify several other classes of box-perfect graphs. We also develop a general and powerful method for establishing box-perfectness.