A note on the geodetic number and the Steiner number of AT-free graphs

Abstract

We study two graph parameters, namely the geodetic number and the Steiner number, which are related to the concept of convexity. We show that, in asteroidal triple-free graphs, the Steiner number is greater than or equal to the geodetic number. This answers a question posed by Hernando, Jiang, Mora, Pelayo, and Seara in 2005. Besides, we show that the gap between the two parameters can be arbitrarily large even in unit-interval graphs, a proper subclass of AT-free graphs.