On the geodetic hull number of Pk-free graphs

Partially answering a question posed by Araujo, Morel, Sampaio, Soares, and Weber, we show that computing the geodetic hull number of a given P 9 -free graph is NP-hard. Similarly, we show that computing the geodetic interval number of a given P 5 -free graph is NP-hard. Furthermore, we show that the geodetic hull number can be computed in polynomial time for { paw , P 5 } -free graphs, triangle-free graphs in which every six vertices induce at most one P 5 , and { C 3 , ź , C k - 2 , P k } -free graphs for every integer k.