There appears to be no standard term for a graph with graph crossing number 1. In particular, the terms "almost planar graph" (e.g., Karpov 2013) and 1-planar graph (e.g., Fabrici and Madaras 2007, Brandenburg 2021) are used in the literature for a different concepts. In this work, the term "singlecross graph" is therefore used to refer to a graph with graph crossing number 1.
A singlecross graph has graph genus 1 and is therefore a toroidal graph, since the single crossing can
be removed by adding a handle. Not every toroidal graph
is singlecross, however; for example, the complete
graphs 
and 
have graph crossing numbers 3 and 9, respectively.
Möbius ladders are singlecross by construction.
Checking if a graph is singlecross is straightforward using the following algorithm (M. Haythorpe, pers. comm., Apr. 16, 2019). First, confirm that the graph
is nonplanar. Then, for all non-adjacent pairs of edges 
and 
, delete the two edges and create a new vertex 
. Finally, check if any one of the four new graphs obtained
from adding any one of the edges 
, 
, 
, and 
is planar. If so, then the original graph is singlecross.
The numbers of singlecross simple graphs on 
nodes are 0, 0, 0, 0, 1, 12, 162, 3183, 74696, 1892122,
... (A307071), and the numbers of connected
graphs are 0, 0, 0, 0, 1, 11, 149, 3008, 71335, 1814021, ... (A307072).
