  The inverse eigenvalue problem studies the possible spectra among matrices
whose off-diagonal entries have their zero-nonzero patterns described by the
adjacency of a graph $G$. In this paper, we refer to the $i$-nullity pair of a
matrix $A$ as $(\operatorname{null}(A), \operatorname{null}(A(i))$, where
$A(i)$ is the matrix obtained from $A$ by removing the $i$-th row and column.
The inverse $i$-nullity pair problem is considered for complete graphs, cycles,
and trees. The strong nullity interlacing property is introduced, and the
corresponding supergraph lemma and decontraction lemma are developed as new
tools for constructing matrices with a given nullity pair.
