I have an answer that should be logically correct but the SQL part is not complete. The idea is based on intersections of adjacent polygons. The intesections are either lines if polygons share segment(s), or points if polygons meet at one vertex.
The first step is to compute the intersection of one polygon and the union of all the polygons on the layer, except the selected polygon.
The intersection of polygon 1 and union of all the rest is a multilinestring with two members. The intersection of polygon 6 and anything else but polygon 6 consists of four linestrings members which are connected.
The second step is to try if it is possible to build a new polygon from the intersections. For the connected lines from polygon 6 it is possible but not for the two linestrings from polygon 1.
This is the SQL that I used for testing.
select ST_MakePolygon(ST_LineMerge(c.intersection)) from
ST_Intersection(sub_a.a,sub_b.b) as intersection from
(select ST_Union("GEOMETRY") as a from outline_test where gid!=6) as sub_a,
(select "GEOMETRY" as b from outline_test where gid=6) as sub_b) as c;
With gid=1 the result is an error:
ERROR: lwpoly_from_lwlines: shell must have at least 4 points
and with gid=2:
ERROR: lwpoly_from_lwlines: shell must be closed