  We obtain upper bounds on the number of nodal domains of Laplace
eigenfunctions on chain domains with Neumann boundary conditions. The chain
domains consist of a family of planar domains, with piecewise smooth boundary,
that are joined by thin necks. Our work does not assume a lower bound on the
width of the necks in the chain domain. As a consequence, we prove an upper
bound on the number of Courant sharp eigenfunctions that is independent of the
widths of the necks.
