  We study how the thermodynamic properties of the Triangular Plaquette Model
(TPM) are influenced by the addition of extra interactions. The thermodynamics
of the original TPM is trivial, while its dynamics is glassy, as usual in
Kinetically Constrained Models. As soon as we generalize the model to include
additional interactions, a thermodynamic phase transition appears in the
system. The additional interactions we consider are either short ranged,
forming a regular lattice in the plane, or long ranged of the small-world kind.
In the case of long-range interactions we call the new model Random-Diluted
TPM. We provide arguments that the model so modified should undergo a
thermodynamic phase transition, and that in the long-range case this is a glass
transition of the "Random First-Order" kind. Finally, we give support to our
conjectures studying the finite temperature phase diagram of the Random-Diluted
TPM in the Bethe approximation. This corresponds to the exact calculation on
the random regular graph, where free-energy and configurational entropy can be
computed by means of the cavity equations.
