#Wolfram Language <s>64</s> 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.

    ""<>#&/@#&/@Select[#~Subsets~{2},IntersectingQ@@#&]&


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