Equivalently, is there a graph that contains an infinite simple path that has a start and an end point? My intuition is that there is no such graph, but I'm finding it hard to articulate why.



Is it possible to define a graph that has two vertices that are infinitely far apart?
One reasonable definition of "simple path" is a connected acyclic subgraph with vertex degree at most two. But, to avoid circularity, we need a definition for being connected. I think the standard definition is that a subgraph is connected iff every two vertices are at a finite distance from each other, so with this definition the answer is trivially no.


On the other hand, if you define a "simple path" to be an acyclic graph in which all vertices have degree one or two and at most two of them have degree one (not the definition I'd choose, but it works for finite graphs) then with this definition the answer is yes: remove a single vertex from the middle of the double-ended infinite path indexed by the integers.


So the answer to your question depends on the details of your definitions.


