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Maximal NIC-plane graphs
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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).
World Scientific Pub Co Pte Ltd
Title: Maximal NIC-plane graphs
Description:
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).
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