An overview is given of the logarithmic minimal models as prototypical examples of logarithmic CFTs.  In the boundary theory, multiplication in the Grothendieck ring of W-projective representations leads to a Verlinde-like formula involving A-type twisted affine graphs and their coset graphs.
This provides compact formulas for the conformal partition functions with W-projective boundary conditions. On the torus, modular invariant partition functions are proposed as sesquilinear forms in W-projective and rational minimal characters that are encoded by the same coset fusion graphs.