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Edge contraction is a fundamental operation in graph theory that involves modifying a given graph by merging two adjacent vertices into a single vertex. This operation is often used in various algorithms and theoretical contexts, such as simplifying graphs, network flow problems, and studying graph properties. Here's how edge contraction works: 1. **Consider an edge** \(e\) between two vertices \(u\) and \(v\) in a graph \(G\).