Maximal NIC-plane graphs

A graph is called NIC-planar if it admits a drawing in the plane such that each edge is crossed at most once and two pairs of crossing edges share at most one vertex. A graph together with a NIC-planar drawing is a NIC-plane graph. A graph is maximal NIC-planar or NIC-plane, if we cannot add any missing edge so that the resulting graph is still NIC-planar or NIC-plane. A vertex in a graph [Formula: see text] is called a dominating vertex if it is adjacent to all other vertices in [Formula: see text]. In this paper, we show that a maximal NIC-plane graph of order [Formula: see text] with a dominating vertex has at least [Formula: see text] edges and this bound is tight, and this also answers a question posed by Ouyang et al. (Appl. Math. Comput. 362 (2019) 124537).