GEODETIC NUMBER VERSUS HULL NUMBER IN P3-CONVEXITY

We study the graphs G for which the hull number h(G) and the geodetic number g(G) with respect to P-3-convexity coincide. These two parameters correspond to the minimum cardinality of a set U of vertices of G such that the simple expansion process which iteratively adds to U all vertices outside of U having two neighbors in U produces the whole vertex set of G either eventually or after one iteration, respectively. We establish numerous structural properties of the graphs G with h(G) = g(G), allowing for the constructive characterization as well as the efficient recognition of all such graphs that are triangle-free. Furthermore, we characterize-in terms of forbidden induced subgraphs-the graphs G that satisfy h(G') = g(G') for every induced subgraph G' of G.