In general, a structure is called rigid if it admits only the trivial structure preserving transformations. Of course, what is trivial depends on the context. In [7] the authors understand by a rigid graph one which has the property that the only edge preserving transformation of the vertices is the identity map. In other contexts, however, it is convenient to regard the constant maps as trivial also. (See [2] where a topological space is given which admits as continuous transformations only the identity and the constants.) The purpose of this note is to construct graphs rigid in the later sense; we call them completely rigid.