A 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing point, where it crosses a single additional edge. 1 planar graph

I read the following paper, which stated the following fact about the rectilinear drawing of a 1-plane graph.

**Thomassen, C. (1988), Rectilinear drawings of graphs. J. Graph Theory, 12: 335-341.**https://doi.org/10.1002/jgt.3190120306

**Theorem 3.1.** Let $G$ be a graph represented as a normal drawing of index $0$ or $1$. Then there exists a rectilinear drawing equivalent to $G$ if and only if $G$ contains no $B-$ or $W-$configuration.

From this theorem we know that a 1-planar graph satisfies **any 1-plane drawing** of it without $B-$ and $W-$configuration, then it has no rectilinear drawing.

But I haven't constructed a 1-planar graph without rectilinear drawing so far. It is even doubtful that it exists. This is one of my confusing points. Any advice would be helpful and thanks in advance!

**Some concepts are added below:**

Every finite graph can be drawn in the Euclidean plane such that

**any two edges have at most one point in common**(in particular,**two edges with a common end have no other point in common**), and such that**every point that is not a vertex**and that is on two edges is on precisely two edges and is a cross-point for these two edges. We call such a**drawing normal**.The

**maximum number of crosspoints on an edge**is called the**cross-index**of the drawing.