In solid modeling and computer-aided design, the Euler operators modify the graph of connections to add or remove details of a mesh while preserving its topology.
The boundary representation for a solid object, its surface, is a polygon mesh of vertices, edges and faces.
Then, to denote a meaningful geometric object, the mesh must satisfy the generalized Euler–Poincaré formula The Euler operators preserve this characteristic.
The Eastman paper lists the following basic operators, and their effects on the various terms: Euler operators modify the mesh's graph creating or removing faces, edges and vertices according to simple rules while preserving the overall topology thus maintaining a valid boundary (i.e. not introducing holes).
The operators themselves don't define how geometric or graphical attributes map to the new graph: e.g. position, gradient, uv texture coordinate, these will depend on the particular implementation.