Maximal 1-plane graphs with dominating vertices

•We observed that any maximal 1-plane graph with at most 6 dominating vertices. Then they provided a nice characterization for the structure for any maximal 1-plane graph with k dominating vertices for k = 3; 4; 5; 6.•For the maximal 1-plane graphs with 2 dominating vertices, we obtained tight lower bounds for the number of edges of G.•In addition, we present similar results for maximal 1-plane graphs with 1 dominating vertex. The new (nice) results here settle the open problem of Ouyang et al. in [Applied Mathematics and Computation 362:124537, 2019].