A defense mechanism in epithelial tissue can mechanically eliminate abnormal cells by contracting the cell boundary or area via actomyosin activity. From numerical simulations of the vertex dynamics model and approximate analytical solutions based on deviations from the ground state, here we derived general conditions for mechanical cell elimination (MCE) occurring via cell contraction. In particular, we found that MCE is realized by saddle-node bifurcation in a wide parameter range, and that the size of the eliminated cell is almost constant at the bifurcation point, suggesting the existence of an intrinsic threshold cell area for MCE.