The term "singlecross graph" is used in this work to refer to a graph with graph crossing number 1. 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 2014) and 1-planar graph (e.g., Fabrici and Madaras 2007, Brandenburg 2024) are used in the literature for different concepts.
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 on 
vertices, with 
, 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. For each pair of edges 
and 
with four distinct endpoints, delete both edges
and add a new vertex 
joined to all four endpoints by the edges 
, 
, 
, and 
. The original graph
is singlecross iff at least one of the resulting graphs
is planar.
The numbers of singlecross simple graphs on 
, 2, ... nodes are
0, 0, 0, 0, 1, 12, 162, 3183, 74696, 1892122, ... (OEIS A307071),
and the numbers of connected graphs are 0, 0,
0, 0, 1, 11, 149, 3008, 71335, 1814021, ... (OEIS A307072).
