Convexity, geodetic, and hull numbers of the join of graphs

In this paper, we characterize the convex sets in the join of two graphs in a more general setting and determine its convexity number. We also show that a result in [1] concerning the geodetic number of the join of graphs does not always hold. In particular, we show that the geodetic number of the join of any two connected non-complete graphs is either 2, 3 or 4. Further, we characterize those joins which yield geodetic number equal to 2 and those with geodetic number equal to 3. Finally, we define and use the concept of 2-path closure absorbing set in a graph to characterize the hull sets in G + Km. We improve and correct a result in [5] by obtaining a more elegant expression for the hull number of the join G + Km̈, where G is a connected non-complete graph