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Encontrando a floresta abrangente (COM RECURSIVO, PostgreSQL 9.5)


Eu escrevi uma resposta para uma pergunta semelhante há algum tempo:Como encontrar todos os subgrafos conectados de um gráfico não direcionado . Nessa questão eu usei o SQL Server. Veja essa resposta para uma explicação detalhada dos CTEs intermediários. Eu adaptei essa consulta ao Postgres.

Pode ser escrito de forma mais eficiente usando o recurso de matriz Postgres em vez de concatenar o caminho em um text coluna.
WITH RECURSIVE
CTE_Idents
AS
(
    SELECT old AS Ident
    FROM identities

    UNION

    SELECT new AS Ident
    FROM identities
)
,CTE_Pairs
AS
(
    SELECT old AS Ident1, new AS Ident2
    FROM identities
    WHERE old <> new

    UNION

    SELECT new AS Ident1, old AS Ident2
    FROM identities
    WHERE old <> new
)
,CTE_Recursive
AS
(
    SELECT
        CTE_Idents.Ident AS AnchorIdent 
        , Ident1
        , Ident2
        , ',' || Ident1 || ',' || Ident2 || ',' AS IdentPath
        , 1 AS Lvl
    FROM 
        CTE_Pairs
        INNER JOIN CTE_Idents ON CTE_Idents.Ident = CTE_Pairs.Ident1

    UNION ALL

    SELECT 
        CTE_Recursive.AnchorIdent 
        , CTE_Pairs.Ident1
        , CTE_Pairs.Ident2
        , CTE_Recursive.IdentPath || CTE_Pairs.Ident2 || ',' AS IdentPath
        , CTE_Recursive.Lvl + 1 AS Lvl
    FROM
        CTE_Pairs
        INNER JOIN CTE_Recursive ON CTE_Recursive.Ident2 = CTE_Pairs.Ident1
    WHERE
        CTE_Recursive.IdentPath NOT LIKE ('%,' || CTE_Pairs.Ident2 || ',%')
)
,CTE_RecursionResult
AS
(
    SELECT AnchorIdent, Ident1, Ident2
    FROM CTE_Recursive
)
,CTE_CleanResult
AS
(
    SELECT AnchorIdent, Ident1 AS Ident
    FROM CTE_RecursionResult

    UNION

    SELECT AnchorIdent, Ident2 AS Ident
    FROM CTE_RecursionResult
)
,CTE_Groups
AS
(
  SELECT
    CTE_Idents.Ident
    ,array_agg(COALESCE(CTE_CleanResult.Ident, CTE_Idents.Ident) 
        ORDER BY COALESCE(CTE_CleanResult.Ident, CTE_Idents.Ident)) AS AllIdents
  FROM
    CTE_Idents
    LEFT JOIN CTE_CleanResult ON CTE_CleanResult.AnchorIdent = CTE_Idents.Ident
  GROUP BY CTE_Idents.Ident
)
SELECT AllIdents
FROM CTE_Groups
GROUP BY AllIdents
;

Eu adicionei uma linha (7,X,Y) aos seus dados de amostra.

SQL Fiddle

Resultado
|          allidents |
|--------------------|
|   Lydia,Mary,Nancy |
| Albert,Bob,Charles |
|                X,Y |
|                Zoe |