Spanning plane subgraphs of $1$-plane graphs

A graph drawn on the plane is called $1$-plane if each edge is crossed at most once by another edge. In this paper, we show that every $4$-connected $1$-plane graph has a connected spanning plane subgraph. We also show that there exist infinitely many $4$-connected $1$-plane graphs that have no $2$-connected spanning plane subgraphs. Moreover, we consider the condition of $k$ and $l$ such that every $k$-connected $1$-plane graph has an $l$-connected spanning plane subgraph.