A note on the convexity number of complementary prisms

In the geodetic convexity, a set of vertices S of a graph G is convex if all vertices belonging to any shortest path between two vertices of S lie in S. The cardinality con(G) of a maximum proper convex set S of G is the convexity number of G. The complementary prism GG of a graph G arises from the disjoint union of the graphs G and G by adding the edges of a perfect matching between the corresponding vertices of G and G. In this work, we prove that the decision problem related to the convexity number is NP-complete even restricted to complementary prisms, we determine con(GG) when G is disconnected or G is a cograph, and we present a lower bound when diam(G) ≠ 3.