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We develop the theory of transformation semigroups that have degree 2, thatis, act by partial functions on a finite set such that the inverse image ofpoints have at most two elements. We show that the graph of fibers of such anaction gives a deep connection between semigroup theory and graph theory. It isknown that the Krohn-Rhodes complexity of a degree 2 action is at most 2. Weshow that the monoid of continuous maps on a graph is the translational hull ofan appropriate 0-simple semigroup. We show how group mapping semigroups can beconsidered as regular covers of their right letter mapping image and relatethis to their graph of fibers.

Degree 2 Transformation Semigroups as Continuous Maps on Graphs: Foundations and Structure