  The complexity of the matching polytope of graphs may be measured with the
maximum length $\beta$ of a starting sequence of odd ears in an
ear-decomposition. Indeed, a theorem of Edmonds and Pulleyblank shows that its
facets are defined by 2-connected factor-critical graphs, which have an odd
ear-decomposition (according to a theorem of Lov\'asz). In particular,
$\beta(G) \leq 1$ if and only if the matching polytope of the graph $G$ is
completely described by non-negativity, star and odd-circuit inequalities. This
is essentially equivalent to the h-perfection of the line-graph of $G$, as
observed by Cao and Nemhauser.
  The complexity of computing $\beta$ is apparently not known. We show that
deciding whether $\beta(G)\leq 1$ can be executed efficiently by looking at any
ear-decomposition starting with an odd circuit and performing basic modulo-2
computations. Such a greedy-approach is surprising in view of the complexity of
the problem in more special cases by Bruhn and Schaudt, and it is simpler than
using the Parity Minor Algorithm.
  Our results imply a simple polynomial-time algorithm testing h-perfection in
line-graphs (deciding h-perfection is open in general). We also generalize our
approach to binary matroids and show that computing $\beta$ is a
Fixed-Parameter-Tractable problem (FPT).
