Contractible Edges in 2-Connected Locally Finite Graphs

In this paper, we prove that every contraction-critical 2-connected infinite graph has no vertex of finite degree and contains uncountably many ends. Then, by investigating the distribution of contractible edges in a 2-connected locally finite infinite graph G, we show that the closure of the subgraph induced by all the contractible edges in the Freudenthal compactification of G is 2-arc-connected. Finally, we characterize all 2-connected locally finite outerplanar graphs nonisomorphic to K3 as precisely those graphs such that every vertex is incident to exactly two contractible edges as well as those graphs such that every finite bond contains exactly two contractible edges.