Vertex splitting is an operation on a finite simple graph that replaces a graph vertex 
by two adjacent vertices 
and 
and redistributes the edges
formerly incident to 
. Each former neighbor
of 
is joined to at least one of 
and 
, and may be joined to both. All
other edges are unchanged. Thus edge
contraction of 
, with loops
removed and parallel edges identified, recovers
the original graph. This is an inverse contraction of the
kind called edge splitting in this work. Other
uses of that name describe a different operation (Schulze and Whiteley 2017, pp. 1597-1599).
In rigidity theory, for a positive integer 
, a 
-dimensional vertex split has exactly 
common neighbors of the
two new vertices. Whiteley (1990) proved that this
operation preserves generic rigidity in 
. A split is nontrivial when
both new vertex degrees are at least 
.
Cruickshank (2026) reports a proof that, for every integer 
, a nontrivial 
-dimensional vertex split of a generically
globally rigid graph in 
is again generically globally rigid. In contraction notation,
if 
is generically globally rigid, 
, 
, and 
, then 
is generically globally
rigid. A corollary allows at least 
common neighbors. The proof
establishes the stronger assertion that duplicating the position of the contracted
graph vertex in any generic realization of 
gives an infinitesimally rigid framework
of 
with 
and 
coincident. The two velocity vectors are allowed to differ when testing infinitesimal
rigidity.
Cruickshank credits ChatGPT 6.1 Sol with discovering the new proofs and reports checking the arguments. Independent specialist verification or formal certification of the complete proofs has not been reported.
