t-Perfection Is Always Strong for Claw-Free Graphs

A connected graph $G$ is called $t$-perfect if its stable set polytope is determined by the nonnegativity, edge, and odd-cycle inequalities. Moreover, $G$ is called strongly $t$-perfect if this system is totally dual integral. It is an open problem whether $t$-perfection is equivalent to strong $t$-perfection. We prove the equivalence for the class of claw-free graphs.