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.
['Wing-Kai Hon', 'Ton Kloks', 'Hsiang Hsuan Liu', 'Hung-Lung Wang', 'Yue-Li Wang']
Convexities in Some Special Graph Classes ---New Results in AT-free Graphs and Beyond