Here we study the two-dimensional Kaya-Berker model, with a site occupancy p of one sublattice, by using a polynomial-time exact ground-state algorithm. Thus, we were able to obtain T=0 results in exact equilibrium for rather large system sizes up to 777^{2} lattice sites. We obtained the sublattice magnetization and the corresponding Binder parameter. We found a critical point p_{c}=0.64(1) beyond which the sublattice magnetization vanishes. This is clearly smaller than previous results which were obtained by using nonexact approaches for much smaller systems. For each realization we also created minimum-energy domain walls from two ground-state calculations, for periodic and antiperiodic boundary conditions. The analysis of the mean and the variance of the domain-wall energy shows that there is no thermodynamic stable spin-glass phase at nonzero temperature, in contrast to previous claims about this model. For large values of p, the standard deviation of the domain-wall decreases with the system size like a power law with exponent roughly θ≃-0.1, which is different from the standard two-dimensional Ising spin glass where θ≃-0.29.