  We study convexity properties of graphs. In this paper we present a
linear-time algorithm for the geodetic number in tree-cographs. Settling a
10-year-old conjecture, we prove that the Steiner number is at least the
geodetic number in AT-free graphs. Computing a maximal and proper monophonic
set in $\AT$-free graphs is NP-complete. We present polynomial algorithms for
the monophonic number in permutation graphs and the geodetic number in $P_4$-
sparse graphs.
