Yup, I think so, too. I haven't sat down to prove that's homogeneous in the sense I described, but it appears so. (The correct term in graph theory turns out to be "connected-homogeneous" or for short, "C-homogeneous", since the subgraphs H_1 and H_2 below are assumed to be connected.) Another definition of an isomorphic graph G is this, I think: Let Q/Z denote the group of rationals (mod 1). Let G be the graph having Q/Z as vertices with the rational q (mod 1) connected to all other rationals (mod 1) except for q ± 1/3 (mod 1), It turns out that all connected, countable C-homogeneous graphs were classified in 2010 by R. Gray and D. Macpherson in <https://www.sciencedirect.com/science/article/pii/S009589560900046X>. —Dan Keith Lynch wrote: ----- Dan Asimov <dasimov@earthlink.net> wrote:

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.

If I understand you correctly, I think the graph that links each integer n to every other integer except n-1 and n+1 would be a solution. -----