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#Wolfram Language 64 53 bytes

This finds all the Subset's of length 2, Select's those in which the nodes of one pair intersect with the nodes of another pair (indicating that the pairs share a node), and StringJoin's the nodes for selected pairs.

The code employs braces, "{}", as list delimiters, as is customary in Wolfram Language.


1 byte saved thanks to Mr. Xcoder.