  In this paper we revisit an example of Celikbas and Takahashi concerning the
reflexivity of tensor products of modules. We study Tor-rigidity and the
Hochster--Huneke graph with vertices consisting of minimal prime ideals, and
determine a condition with which the aforementioned example cannot occur. Our
result, in particular, corroborates the Second Rigidity Theorem of Huneke and
Wiegand.
