On box-perfect graphs

Let G=(V,E) be a graph and let AG be the clique-vertex incidence matrix of G. It is well known that G is perfect iff the system AGx≤1, x≥0 is totally dual integral (TDI). In 1982, Cameron and Edmonds proposed to call G box-perfect if the system AGx≤1, x≥0 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.