{ {A,B},{B,C},{C,D},{A,D} } is not decomposable.

{ {A,B,C},{A,C,D} } is decomposable.
ABC-ACD is a join forest for it.

{ {TbOrCa, XRay}, {Bronchitis, Smoking}, {Tuberculosis,
VisitAsia}, {Bronchitis, Dyspnea,
TbOrCa}, {Cancer, Smoking}, {Cancer, TbOrCa, Tuberculosis}}
is not decomposable. Graham's algorithm fails to make further progress
once we have: {  {Bronchitis, Smoking}, {Cancer, Smoking},{Bronchitis,
TbOrCa}, {Cancer, TbOrCa }}.

{ {TbOrCa, XRay}, {Bronchitis, Smoking}, {Tuberculosis,
VisitAsia}, {Bronchitis, Dyspnea, TbOrCa}, {Cancer, TbOrCa,
Tuberculosis} } is decomposable.