Tutte's theorem

Tutte's theorem provides a condition to determine if a graph (a set of points connected by lines) can be perfectly matched, meaning every point pairs with exactly one other without leftovers. It states that a graph has such a perfect matching if and only if, for every subset of points removed, the remaining graph’s disconnected parts are limited in a specific way, related to the number of these parts and their isolated points. This theorem helps identify matchings in complex networks, with applications ranging from chemistry to algorithm design.