I just thought of one, so I'll make it a puzzle: Find a connected graph G on a countably infinite set of vertices such that * each vertex has infinite valence * each vertex is non-adjacent to exactly 2 vertices * any isomorphism of a subgraph H_1 to a subgraph H_2 extends to an isomorphism of the entire graph G to itself. Note: By a "subgraph" here is mean any collection of the vertices and any collection of the edges which together form a connected graph. —Dan