Polynomial interpolation with equidistant nodes is notoriously
unreliable due to the Runge phenomenon, and is also numerically
ill-conditioned. By taking advantage of the optimality of the
interpolation processes on Chebyshev nodes, one of the best strategies
to defeat the Runge phenomenon is to use the mock-Chebyshev points,
which are selected from a satisfactory uniform grid, for polynomial
interpolation. Yet, little literature exists on the computation of these
points. In this study, we investigate the properties of the
mock-Chebyshev nodes and propose a subsetting method for constructing
mock-Chebyshev grids. Moreover, we provide a precise formula for the
cardinality of a satisfactory uniform grid. Some numerical experiments
using the points obtained by the method are given to show the
effectiveness of the proposed method and numerical results are also
provided.