Tutte's theorem is a fundamental result in graph theory that provides a criterion for determining whether a graph can be decomposed into a perfect matching. A perfect matching is a set of edges that pairs up all the vertices of the graph without any overlaps. The theorem states that a graph has a perfect matching if and only if, for every subset of its vertices, the number of odd-sized components in the subgraph formed by removing that subset is at most equal to the number of vertices removed. This theorem is named after the mathematician William Tutte, who made significant contributions to the field of combinatorial mathematics. Tutte's theorem has important applications in various areas, including network design and optimization, as it helps in understanding how to efficiently pair elements in a set.