Graphs with convex domination number close to their order

For a connected graph G = (V,E), a set D ⊆ V(G) is a dominating set of G if every vertex in V(G)-D has at least one neighbour in D. The distance ${\displaystyle d_G(u,v)}$ between two vertices u and v is the length of a shortest (u-v) path in G. An (u-v) path of length ${\displaystyle d_G(u,v)}$ is called an (u-v)-geodesic. A set X ⊆ V(G) is convex in G if vertices from all (a-b)-geodesics belong to X for any two vertices a,b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number ${\displaystyle {\gamma }_{con}\left(G\right)}$ of a graph G is the minimum cardinality of a convex dominating set in G. Graphs with the convex domination number close to their order are studied. The convex domination number of a Cartesian product of graphs is also considered.