Let P be a class of graphs; a graph Γ with vertex set V is locally P-homogeneous if whenever U ⊆ V and the vertex subgraph (U) lies in P, then each automorphism of (U) extends to an automorphism of Γ. Let C be the class of connected graphs, Q the class of cones, R the class of “rakes”; we classify locally finite, locally C-homogeneous graphs, and prove that a locally finite, locally (Q ⌣ R)-homogeneous graph is either locally C-homogeneous, or is the Levi graph of the sevenpoint projective plane.
