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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
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