  This paper presents a Meyer-Vietoris type gluing formula for a conformal
invariant of a Riemannian surface with boundary that is defined by the
determinant of the Dirichlet-to-Neumann operator. The formula is used to bound
the asymptotics of the invariant under degeneration. It is shown that the
associated height function on the moduli space of hyperbolic surfaces with
geodesic boundary is proper only in genus zero. Properness implies a
compactness theorem for Steklov isospectral metrics in the case of genus zero.
The formula also provides asymptotics for the determinant of the Laplacian with
Dirichlet or Neumann boundary conditions. For the proof, we derive an extension
of Kirchhoff's weighted matrix tree theorem for graph Laplacians with an
external potential.
