The density of maximal IC-plane graphs and maximal NIC-plane graphs

In this paper, we show that any maximal IC-plane graph of order $n$ has at least $\lceil \frac{7}{3}n-\frac{14}{3}\rceil$ edges, and any maximal NIC-plane graph of order $n$ has at least $\lceil \frac{11}{5}n-\frac{18}{5}\rceil$ edges. Moreover, we show that both results are tight for infinitely many integers $n$.