  Majorana networks, whose vertices represent localized Majorana modes and
edges correspond to bilinear mixing amplitudes between them, provide a unified
framework for describing the low energy physics of several interesting systems.
Such networks are known to exhibit topologically protected collective Majorana
modes if the combinatorial problem of maximum matchings (maximally-packed dimer
covers) of the underlying graph has unmatched vertices (monomers), as is
typically the case if the network is disordered. These collective Majorana
modes live in ``${\mathcal R}$-type regions'' of the disordered graph, which
host the unmatched vertices (monomers) in any maximum matching and can be
identified using the graph theoretical Gallai-Edmonds decomposition. Here, we
focus on vacancy disorder (site dilution) in general (nonbipartite) two
dimensional lattices such as the triangular and Shastry-Sutherland lattices,
and study the random geometry of such ${\mathcal R}$-type regions and their
complements, i.e., ``${\mathcal P}$-type regions'' from which monomers are
excluded in any maximum matching of the lattice. These ${\mathcal R}$-type and
${\mathcal P}$-type regions are found to display a sharply-defined {\em
Gallai-Edmonds percolation} transition at a critical vacancy density $n_v^{\rm
crit}$ that lies well within the geometrically percolated phase of the
underlying disordered lattice. For $n_v<n_v^{\rm crit}$, ${\mathcal R}$-type
regions percolate but there is a striking lack of self-averaging even in the
thermodynamic limit, with the ensemble average being macroscopically different
from the properties of individual samples: Each sample has exactly one infinite
cluster, which is of type ${\mathcal R}$ ($\mathcal P$) in a weakly
$n_v$-dependent fraction $f <1$ ($1-f < 1$) of the samples.
