  We study the effects of a domain deformation to the nodal set of Laplacian
eigenfunctions when the eigenvalue is degenerate. In particular, we study
deformations of a rectangle that perturb one side and how they change the nodal
sets corresponding to an eigenvalue of multiplicity 2. We establish geometric
properties, such as number of nodal domains, presence of crossings, and
boundary intersections, of nodal sets for a large class of boundary
deformations and study how these properties change along each eigenvalue branch
for small perturbations. We show that internal crossings of the nodal set break
under generic deformations and obtain estimates on the location and regularity
of the nodal sets on the perturbed rectangle.
