Geodetic Convexity Parameters for Graphs with Few Short Induced Paths

We study several parameters of geodetic convexity for graph classes defined by restrictions concerning short induced paths. Partially answering a question posed by Araujo et al., 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. On the positive side, we identify several graph classes for which the geodetic hull number can be computed efficiently. Furthermore, following a suggestion of Campos et al., we show that the geodetic interval number, the geodetic convexity number, the geodetic Caratheodory number, and the geodetic Radon number can all be computed in polynomial time for $$q,q-4$$-graphs.

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