Definition:Edge Contraction

Definition
Let $G$ be an undirected graph.

Let $e \in E\left({G}\right)$ be an edge of $G$.

Then the graph obtained by contracting $e$ in $G$, denoted by $G / e$, is the graph $H$ defined by:
 * $V\left({H}\right) = \left({V \left({G}\right) \setminus e}\right) \cup \left\{{v}\right\}$
 * $E\left({H}\right) = \left\{{f \in E \left({G}\right) : f \cap e = \varnothing}\right\} \cup \left\{{u v : \exists f \in E\left({G}\right) : u \in f \setminus e, f \cap e \ne \varnothing}\right\}$

where $v \not\in V\left({G}\right)$ is a fresh object.

Informally, it is the graph obtained from $G$ by replacing the vertices incident to $e$ with a single vertex adjacent to all their neighbors.

Also see

 * Edge Deletion, which is the removal of an edge.
 * Graph Minor, which is a graph obtained from another by contraction, edge deletion, or isolated vertex deletion.