  Given a topology of local parity-check constraints, a maximally recoverable
code (MRC) can correct all erasure patterns that are information-theoretically
correctable. In a grid-like topology, there are $a$ local constraints in every
column forming a column code, $b$ local constraints in every row forming a row
code, and $h$ global constraints in an $(m \times n)$ grid of codeword.
Recently, Gopalan et al. initiated the study of MRCs under grid-like topology,
and derived a necessary and sufficient condition, termed as the regularity
condition, for an erasure pattern to be recoverable when $a=1, h=0$.
  In this paper, we consider MRCs for product topology ($h=0$). First, we
construct a certain bipartite graph based on the erasure pattern satisfying the
regularity condition for product topology (any $a, b$, $h=0$) and show that
there exists a complete matching in this graph. We then present an alternate
direct proof of the sufficient condition when $a=1, h=0$. We later extend our
technique to study the topology for $a=2, h=0$, and characterize a subset of
recoverable erasure patterns in that case. For both $a=1, 2$, our method of
proof is uniform, i.e., by constructing tensor product $G_{\text{col}} \otimes
G_{\text{row}}$ of generator matrices of column and row codes such that certain
square sub-matrices retain full rank. The full-rank condition is proved by
resorting to the matching identified earlier and also another set of matchings
in erasure sub-patterns.
