  McCuaig (2001, Brace Generation, J. Graph Theory 38: 124-169) proved a
generation theorem for braces, and used it as the principal induction tool to
obtain a structural characterization of Pfaffian braces (2004, P{\'o}lya's
Permanent Problem, Electronic J. Combinatorics 11: R79).
  A brace is minimal if deleting any edge results in a graph that is not a
brace. From McCuaig's brace generation theorem, we derive our main theorem that
may be viewed as an induction tool for minimal braces. As an application, we
prove that a minimal brace of order $2n$ has size at most $5n-10$, when $n \geq
6$, and we provide a complete characterization of minimal braces that meet this
upper bound.
  A similar work has already been done in the context of minimal bricks by
Norine and Thomas (2006, Minimal Bricks, J. Combin. Theory Ser. B 96: 505-513)
wherein they deduce the main result from the brick generation theorem due to
the same authors (2007, Generating Bricks, J. Combin. Theory Ser. B 97:
769-817).

Combinatorics