When is it true that if $G$ is isomorphic to a spanning subgraph of $H$ and $H$ is isomorphic to a spanning subgraph of $G$, then $G$ is isomorphic to $H$?

Clearly this is true if $G$ and $H$ are finite graphs; however, this is not necessarily true for infinite graphs. For instance, let $G$ be an infinite clique together with infinitely many isolated vertices and let $H$ be two disjoint infinite cliques together with infinitely many isolated vertices. We posed this problem in Corsten, DeBiasio, and McKenney - Density of monochromatic infinite subgraphs II (see Problem 2.12), but since this question seems more basic and is only tangential to our results, I figured I would ask here as well.

Addendum 1: After doing some more digging, I found this related post Non-isomorphic graphs with bijective graph homomorphisms in both directions between them which just asked for examples of such graphs $G$ and $H$ where $G$ and $H$ are not isomorphic.

Addendum 2: There was a comment yesterday, which for some reason seems to have been deleted, suggesting the term "co-hopfian graph." I found this paper Cain and Maltcev - Hopfian and co-hopfian subsemigroups and extensions which defines co-hopfian graphs (see the paragraph before Lemma 4.5) as those in which every *injective* homomorphism from $G$ to $G$ (i.e. injective endomorphism) is an isomorphism. It's unclear to me if this makes a difference in the characterization, but I now believe my question is equivalent to "Which graphs $G$ have the property that every *bijective* homomorphism from $G$ to $G$ (i.e. bijective endomorphism) is an automorphism." Sorry to overdo it, but my original question has now become three questions:

Which graphs $G$ have the property that every

*injective*endomorphism is an automorphism? (equivalently, when is it true that if $G$ is isomorphic to a subgraph of $H$ and $H$ is isomorphic to a subgraph of $G$, then $G$ is isomorphic to $H$?)Which graphs $G$ have the property that every

*bijective*endomorphism is an automorphism? (equivalently, when is it true that if $G$ is isomorphic to a*spanning*subgraph of $H$ and $H$ is isomorphic to a*spanning*subgraph of $G$, then $G$ is isomorphic to $H$?)Are the answers to 1 and 2 the same?