I had a question which asked whether 2-regular graphs have any articulation points. We assumed finite graphs so it's just a disjoint union of cycles. However if we allow infinite graphs how do we approach an infinite chain? Is it the same as an infinite cycle?
I was thinking along the lines of totally ordered sets. Does the ring of integers contain articulation points?
A cycle is defined as a closed walk without repeated vertices. With this definition, an infinite chain cannot be a cycle, because a walk from any starting vertex cannot return to where it started without backtracking on itself – you cannot "go to infinity and back again". It follows that every vertex in an infinite chain is an articulation point.