# Are t-perfect graphs strongly t-perfect?

A graph *G=(V, E)* is t-perfect if its stable set polytope is determined by the system

[math]\begin{array}{rll} x(v) & \geq 0 & \text{ for each } v \in V,\\ x(u) + x(v) & \leq 1 & \text{ for each } uv \in E,\\ x(V(C)) & \leq \lfloor |V(C)|/2 \rfloor & \text{ for each odd circuit } C. \end{array} [/math]

A graph is strongly t-perfect if the above system is TDI.

Is it true that every t-perfect graph is strongly t-perfect?

## Remarks

Gerards ^{[1]} proved that a graph without an odd [math]K_4[/math]-subdivision is strongly t-perfect. This has been generalized by Schrijver ^{[2]} to all graphs without a bad subdivision of *K _{4}* (a subdivision of

*K*is

_{4}**bad**if it is not t-perfect).

Bruhn and Stein ^{[3]} proved that claw-free t-perfect graphs are strongly t-perfect.

## References

- ↑ A. M. Gerards,
*A min-max relation for stable sets in graphs with no odd-K*, DOI link_{4} - ↑ A. Schrijver,
*Strong t-perfection of bad-K*, DOI link, Author link._{4}-free graphs - ↑ H. Bruhn, M. Stein,
*t-perfection is always strong for claw-free graphs*, DOI link, author link