  A graph is a core or unretractive if all its endomorphisms are automorphisms.
Well-known examples of cores include the Petersen graph and the graph of the
dodecahedron -- both generalized Petersen graphs. We characterize the
generalized Petersen graphs that are cores. A simple characterization of
endomorphism-transitive generalized Petersen graphs follows. This extends the
characterization of vertex-transitive generalized Petersen graphs due to
Frucht, Graver, and Watkins and solves a problem of Fan and Xie.
  Moreover, we study generalized Petersen graphs that are (underlying graphs
of) Cayley graphs of monoids. We show that this is the case for the Petersen
graph, answering a recent mathoverflow question, for the Desargues graphs, and
for the dodecahedron -- answering a question of Knauer and Knauer. Moreover, we
characterize the infinite family of generalized Petersen graphs that are Cayley
graph of a monoid with generating connection set of size two. This extends
Nedela and \v{S}koviera's characterization of generalized Petersen graphs that
are group Cayley graphs and complements results of Hao, Gao, and Luo.
