Degree 2 transformation semigroups as continuous maps on graphs: Complexity and examples

In this paper, we give a number of illuminating examples of transformation semigroups of degree 2 acting on graphs by functions that preserve vertices and edges by inverse image. It is known that the complexity of such a transformation semigroup is at most 2. We give examples that use sophisticated lower bounds to complexity to distinguish between complexity 1 and complexity 2.