  The classical P\'olya-Tchebotarev problem, commonly stated as a max-min
logarithmic energy problem, asks for finding a compact of minimal capacity in
the complex plane which connects a prescribed collection of fixed points.
Variants of this problem have found ramifications and applications in the
theory of non-hermitian orthogonal polynomials, random matrices, approximation
theory, among others. Here we consider an extension of this classical problem,
including a semiclassical external field, and enforcing finitely many
prescribed collections of points to be connected, possibly also to infinity.
Our method is based on Rakhmanov's approach to max-min problems in logarithmic
potential theory, utilizes the developed machinery by Mart\'inez-Finkelshtein
and Rakhmanov on critical measures, and extends the development of Kuijlaars
and the second named author from the context of polynomial external fields to
the semiclassical case considered here.
