[[File:Cuboctahedron.jpg|thumb|The [[cuboctahedron]], a planar locally linear graph that can be formed as the line graph of a cube or by gluing antiprisms onto the inside and outside faces of a 4-cycle]]
A more complicated expansion process applies to [[planar graph]]s. Let <math>G</math> be a planar graph embedded in the plane in such a way that every face is a quadrilateral, such as the graph of a cube. ItGluing followsa [[square antiprism]] onto each face of <math>G</math>, and then deleting the original edges of <math>G</math>, produces a new locally linear planar graph. The numbers of edges and vertices of the result can be calculated from [[Euler's polyhedral formula]] that: if <math>G</math> has <math>n</math> vertices, it has exactly <math>n-2</math> faces., Gluingand athe [[square antiprism]] onto each faceresult of <math>G</math>, and then deletingreplacing the original edgesfaces of <math>G</math>,producesby aantiprisms new locally linear planar graph withhas <math>5(n-2)+2</math> vertices and <math>12(n-2)</math> edges.{{r|z}} For instance, the cuboctahedron can again be produced in this way, from the two faces (the interior and exterior) of a 4-cycle.
The removed 4-cycle of this construction can be seen on the cuboctahedron as a cycle of four diagonals of its square faces, bisecting the polyhedron.
