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 graph is sometimes written as G/e. (Contrast this with G \setminus e, which means simply removing the edge e without merging its incident vertices.) 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.
Formal definition Let G = (V, E) be a graph (
or directed graph) containing an edge e = (u, v) with u \neq v. Let f be a function that maps every vertex in V \setminus\{u, v\} to itself, and otherwise, maps it to a new vertex w. The contraction of e results in a new graph G' = (V', E'), where V' = (V \setminus\{u, v\})\cup\{w\}, E' = E \setminus \{e\}, and for every x \in V, x' = f(x)\in V' is incident to an edge e' \in E' if and only if, the corresponding edge, e \in E is incident to x in G.
Vertex identification Vertex identification (sometimes called
vertex contraction) removes the restriction that the
contraction must occur over vertices sharing an incident edge. (Thus, edge contraction is a special case of vertex identification.) The operation may occur on any pair (or subset) of vertices in the graph. Edges between two
contracting vertices are sometimes removed. If v and v' are vertices of distinct components of G, then we can create a new graph G' by identifying v and v' in G as a new vertex \textbf{v} in G'. More generally, given a
partition of the vertex set, one can identify vertices in the partition; the resulting graph is known as a
quotient graph.
Vertex cleaving Vertex cleaving, which is the same as vertex splitting, means one vertex is being split into two, where these two new vertices are adjacent to the vertices that the original vertex was adjacent to. This is a reverse operation of vertex identification, although in general for vertex identification, adjacent vertices of the two identified vertices are not the same set.
Path contraction Path contraction occurs upon the set of edges in a
path that
contract to form a single edge between the endpoints of the path. Edges incident to vertices along the path are either eliminated, or arbitrarily (or systematically) connected to one of the endpoints.
Twisting Consider two disjoint graphs G_1 and G_2, where G_1 contains vertices u_1 and v_1 and G_2 contains vertices u_2 and v_2. Suppose we can obtain the graph G by identifying the vertices u_1 of G_1 and u_2 of G_2 as the vertex u of G and identifying the vertices v_1 of G_1 and v_2 of G_2 as the vertex v of G. In a
twisting G' of G with respect to the vertex set \{u, v\}, we identify, instead, u_1 with v_2 and v_1 with u_2.
Repeated contractions Given a finite set of edges, the order in which contractions are performed on a graph does not change the result (up to isomorphism). The result reduces to showing that G / e / (f / e) is isomorphic to G / f / (e / f) for two edges e, f of G. == Applications ==