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 P9P9-free graph is NP-hard. Similarly, we show that computing the geodetic interval number of a given P5P5-free graph is NP-hard. Furthermore, we show that the geodetic hull number can be computed in polynomial time for {paw,P5}{paw,P5}-free graphs, triangle-free graphs in which every six vertices induce at most one P5P5, and {C3,…,Ck−2,Pk}{C3,…,Ck−2,Pk}-free graphs for every integer k.