Goldberg–Coxeter construction

The Goldberg–Coxeter construction or Goldberg–Coxeter operation (GC construction or GC operation) is a graph operation defined on regular polyhedral graphs with degree 3 or 4. It also applies to the dual graph of these graphs, i.e. graphs with triangular or quadrilateral "faces". The GC construction can be thought of as subdividing the faces of a polyhedron with a lattice of triangular, square, or hexagonal polygons, possibly skewed with regards to the original face: it is an extension of concepts introduced by the Goldberg polyhedra and geodesic polyhedra. The GC construction is primarily studied in organic chemistry for its application to fullerenes, but it has been applied to nanoparticles, computer-aided design, basket weaving, and the general study of graph theory and polyhedra.

The Goldberg–Coxeter construction may be denoted as ${\displaystyle GC_{k,\ell }(G_{0})}$, where ${\displaystyle G_{0}}$ is the graph being operated on, ${\displaystyle k}$ and ${\displaystyle l}$ are integers, ${\displaystyle k>0}$, and ${\displaystyle \ell \geq 0}$.