False twin ($N(u)=N(v)$)

Distinct vertices with equal open neighbourhoods are false twins. They are nonadjacent: adjacency would put one vertex in the other's open neighbourhood but not its own. Equal open neighbourhoods define false-twin classes, which are independent sets and have uniform adjacency to every other such class.

A false-twin class is an equivalence class of vertices with identical open neighbourhoods. In edge-triangle symmetrization, merging nonadjacent false-twin classes by cloning the better local score never decreases the objective. A secondary maximization of squared class sizes forces all distinct classes to be adjacent, giving a complete multipartite graph.