A node-to-circle expansion of a graph 
replaces each degree-
vertex 
by a 
-cycle, with the 
th incident edge of 
attached to the 
th vertex of that cycle. In order to be well-defined for an
abstract graph, a cyclic order of the incident edges specifying which edge attaches
to which position on the replacement cycle (i.e., a rotation
system) must also be specified. Given a rotation
system specifying the cyclic ordering of the edges incident to 
, this ordering is preserved by the node-to-circle expansion
(Brandenburg 2020).
The most common case is the cube-connected cycle graph, which is the node-to-circle expansion of a hypercube graph (Leighton 1992, Brandenburg 2020) when vertices are labeled as bitstrings and neighbors are determined by which coordinate flips, thus giving replacement cycle positions that are exactly the coordinate indices.
The node-to-circle expansion of a graph 
may be denoted 
and is a cubic graph with
vertices and 
edges if G has 
edges and no vertices of degree one or two (Brandenburg 2020).
If 
has graph crossing number 
, then 
has crossing number 
(Brandenburg 2020). Similarly, if a rotation
system comes from an embedding of 
, then performing the node-to-circle expansion with that local
cyclic order gives an embedding of the expanded graph on the same surface, so graph genus is preserved for that choice of order. Conversely,
for any chosen order, contracting each replacement cycle back to a single vertex
recovers 
as a graph minor, so the expanded graph can never
have smaller graph genus than 
. Therefore, graph genus is
preserved by the node-to-circle expansion for vertex orders realized by a minimum-genus
embedding of 
(cf. Brandenburg 2020). Howevr, note that for an arbitrary vertex order not compatible
with any minimum-genus embedding, the node-to-circle expansion can have larger genus.
