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 P4- sparse graphs.
@article{arxiv.1509.04944,
title = {Convexities in Some Special Graph Classes ---New Results in AT-free Graphs and Beyond},
author = {Wing-Kai Hon and Ton Kloks and Hsiang-Hsuan Liu and Hung-Lung Wang and Yue-Li Wang},
journal= {arXiv preprint arXiv:1509.04944},
year = {2015}
}