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We study convexity properties of graphs. In this paper we present alinear-time algorithm for the geodetic number in tree-cographs. Settling a10-year-old conjecture, we prove that the Steiner number is at least thegeodetic number in AT-free graphs. Computing a maximal and proper monophonicset in $\AT$-free graphs is NP-complete. We present polynomial algorithms forthe monophonic number in permutation graphs and the geodetic number in $P_4$-sparse graphs.

Convexities in Some Special Graph Classes ---New Results in AT-free Graphs and Beyond