page_title: Edge contraction
section_title: Definition
hierarchical_section_title: Edge contraction / Definition
context_page_description: In graph theory, an edge contraction is an operation which removes an edge from a graph while simultaneously merging the two vertices that it previously joined. Edge contraction is a fundamental operation in the theory of graph minors. Vertex identification is a less restrictive form of this operation.
context_section_description: The edge contraction operation occurs relative to a particular edge, e. The edge e is removed and its two incident vertices, u and v, are merged into a new vertex w, where the edges incident to w each correspond to an edge incident to either u or v. More generally, the operation may be performed on a set of edges by contracting each edge (in any order).
  The resulting induced graph if sometimes written as G / e. (Contrast this with G \ e which means removing the edge e.) 
  As defined below, an edge contraction operation may result in a graph with multiple edges even if the original graph was a simple graph. However, some authors disallow the creation of multiple edges, so that edge contractions performed on simple graphs always produce simple graphs.
caption: Contracting an edge without creating multiple edges.