Sets of Parter vertices which are Parter sets

Given an Hermitian matrix, whose graph is a tree, having a multiple eigenvalue λ, the Parter-Wiener theorem guarantees the existence of principal submatrices for which the multiplicity of λ increases. The vertices of the tree whose removal gives rise to these principal submatrices are called weak Parter vertices and with some additional conditions are called Parter vertices. A set of k Parter vertices whose removal increases the multiplicity of λ by k is called Parter set. As observed by several authors a set of Parter vertices is not necessarily a Parter set. In this paper we prove that if A is a symmetric matrix, whose graph is a tree, and λ is an eigenvalue of A whose multiplicity does not exceed 3, then every set of Parter vertices, for λ relative to A, is also a Parter set.