A minimally curvy graph is a curvy graph 
none of whose proper topological
minors is a curvy graph. Since 
for every graph 
, this is equivalent to requiring 
for every proper topological
minor 
of 
.
In particular, a graph subdivision that is a
curvy graph is not minimally curvy if it subdivides
another curvy graph.
Since no graph of graph order less than 8 is a curvy graph, a curvy graph of graph order 8 is minimally curvy iff it contains no other curvy graph of graph order 8 as a proper subgraph. The 16-cell graph and 8-double-toroidal graph 8 have no proper subgraphs that are curvy graphs.
Minimal crossing and rectilinear crossing embeddings for these two graphs are illustrated above. The 16-cell
graph 
has 
,
while 8-double-toroidal graph 8 has 
.
A complete multipartite graph 
contains the 16-cell
graph 
as a subgraph iff
 |
Necessity follows because each part can contribute at most two vertices to such a subgraph. Conversely, choose eight vertices,
at most two from each part. Each pair chosen from the same part forms a part of 
. Pair the singly chosen vertices and delete the matching
joining those pairs. Thus every complete
multipartite graph satisfying this inequality
that is a curvy graph, other than the 16-cell
graph itself, is not a minimally curvy graph. This includes 
and 
. Consequently, up to graph
isomorphism, the only minimally curvy graphs of graph
order 8 are the 16-cell graph and 8-double-toroidal
graph 8.
