A weak box-perfect graph theorem

A graph G is called perfect if $\omega \left(H\right)=\chi \left(H\right)$ for every induced subgraph H of G, where $\omega \left(H\right)$ is the clique number of H and $\chi \left(H\right)$ its chromatic number. The Weak Perfect Graph Theorem of Lovász states that a graph G is perfect if and only if its complement $\bar{G}$ is perfect. This does not hold for box-perfect graphs, which are the perfect graphs whose stable set polytope is box-totally dual integral.

We prove that both G and $\bar{G}$ are box-perfect if and only if ${\bar{G}}^{+}$ is box-perfect, where $G^{+}$ is obtained by adding a universal vertex to G. Consequently, $G^{+}$ is box-perfect if and only if ${\bar{G}}^{+}$ is box-perfect. As a corollary, we characterize when the complete join of two graphs is box-perfect.