  Consider the problem, usually called the P\'olya-Chebotarev problem, of
finding a continuum in the complex plane including some given points such that
the logarithmic capacity of this continuum is minimal. We prove that each
connected inverse image $\T_n^{-1}([-1,1])$ of a polynomial $\T_n$ is always
the solution of a certain P\'olya-Chebotarev problem. By solving a nonlinear
system of equations for the zeros of $\T_n^2-1$, we are able to construct
polynomials $\T_n$ with a connected inverse image.
