  Inspired by applications of perfect graphs in combinatorial optimization,
Chv\'{a}tal defined t-perfect graphs in 1970s. The long efforts of
characterizing t-perfect graphs started immediately, but embarrassingly, even a
working conjecture on it is still missing after nearly 50 years. Unlike perfect
graphs, t-perfect graphs are not closed under substitution or complementation.
A full characterization of t-perfection with respect to substitution has been
obtained by Benchetrit in his Ph.D. thesis. Through the present work we attempt
to understand t-perfection with respect to complementation. In particular, we
show that there are only five pairs of graphs such that both the graphs and
their complements are minimally t-imperfect.
