5 of 6 added 110 characters in body

Wolfram Language 64 53 bytes

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

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

The code is especially difficult to read because it employs 4 nested pure (aka "anonymous") functions.

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

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