  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.
