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Abstract
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 $K_3$ 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.
Highlights
We prove that every contraction-critical 2-connected infinite graph has no vertex of finite degree and contains uncountably many ends
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
Since the pioneering work of Tutte [11] who proved that every 3-connected finite graph nonisomorphic to K4 contains a contractible edge, a lot of research has been done on contractible edges in finite graphs
Summary
Since the pioneering work of Tutte [11] who proved that every 3-connected finite graph nonisomorphic to K4 contains a contractible edge, a lot of research has been done on contractible edges in finite graphs. When extending to locally finite infinite graphs, we need the nontrivial statement that if G is a 2-connected locally finite infinite graph such that every vertex is incident to exactly two contractible edges, GC is a Hamilton circle. We will use it to prove an infinite analog of Theorem 2 for any 2-connected locally finite graph G nonisomorphic to K3. We will show that G is outerplanar if and only if every finite bond of G contains exactly two contractible edges
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