  For a given graph $\mathcal{G}$ of order $n$ with $m$ edges, and a real
symmetric matrix associated to the graph,
$M\left(\mathcal{G}\right)\in\mathbb{R}^{n\times n}$, the interlacing graph
reduction problem is to find a graph $\mathcal{G}_{r}$ of order $r<n$ such that
the eigenvalues of $M\left(\mathcal{G}_{r}\right)$ interlace the eigenvalues of
$M\left(\mathcal{G}\right)$. Graph contractions over partitions of the vertices
are widely used as a combinatorial graph reduction tool. In this study, we
derive a graph reduction interlacing theorem based on subspace mappings and the
minmax theory. We then define a class of edge-matching graph contractions and
show how two types of edge-matching contractions provide Laplacian and
normalized Laplacian interlacing. An $\mathcal{O}\left(mn\right)$ algorithm is
provided for finding a normalized Laplacian interlacing contraction and an
$\mathcal{O}\left(n^{2}+nm\right)$ algorithm is provided for finding a
Laplacian interlacing contraction.
