  One-dimensional edges of classical systems in two dimension sometimes show
surprisingly rich phase transitions and critical phenomena, particularly when
the bulk is at criticality. As such a model, we study the surface critical
behavior of the 3-state dilute Potts model whose bulk is tuned at the
tricritical point. To investigate it more precisely than in the previous works
[Y. Deng and H. W. J. Bl\"ote, Phys. Rev. E 70, 035107(R) (2004); Phys. Rev. E
71, 026109 (2005)], we analyze it from the viewpoint of the boundary conformal
field theory (BCFT). The complete classification of the conformal boundary
conditions for the minimal BCFTs discussed in [R. E. Behrend et al., Nucl.
Phys. B 579, 707 (2000)] allows us to collect the twelve boundary fixed points
in the tricritical 3-state Potts BCFT. Employing the tensor network
renormalization method, we numerically study the surface phase diagram of the
tricritical 3-state Potts model in detail, and reveal that the eleven boundary
fixed points among the twelve can be realized on the lattice by controlling the
external field and coupling strength at the boundary. The last unfound fixed
point would be out of the physically sound region in the parameter space,
similarly to the `new' boundary condition in the 3-state Potts BCFT.
