# Cardinality of the set of graphs on a infinite set of vertices

How might one label an infinite graph (which edges can cross over and all nodes are connected to at least 2 edges, such that there are no "dangling lines") to show that there are countably many such graphs, up to isomorphism/homomorphism? Thanks.

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I see that the "counting" tag automatically changes to "combinatorics"...? –  Marc Aug 19 '11 at 16:22
Surely there are uncountably many? I think it's pretty straightforward to construct an injection from $2^\mathbb{N}$ to such graphs. –  Peter Taylor Aug 19 '11 at 17:53