Abstract
A graph G is called 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. A graph, together with a 1-planar drawing is called 1-plane. A graph is maximal 1- planar (1-plane), if we cannot add any missing edge so that the resulting graph is still 1-planar (1-plane). Let G be a maximal 1-plane graph of order n and k dominating vertices. In this paper, we first provide a complete characterization on G for 3≤k≤6. Then we prove that G has at least 52n−4 edges for k=2, and this bound is tight. Finally, we obtain that G has at least 73n−103 edges for k=1. Our results also settle an open problem posed by Ouyang et al. in [Applied Mathematics and Computation 362:124537, 2019].
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