 Edge contraction

An edge contraction merges the two endpoints of an existing edge. For a finite simple graph $G$ and $e=uv$, replace $u,v$ by a new vertex $w$. Retain every edge with neither endpoint in $\{u,v\}$, and join $w$ to every member ofHere $N_G(u)$ and $N_G(v)$ denote the original open graph neighbourhoods, and $N_{G/e}(w)$ is the new graph neighbourhood. Remove the self-loop arising from $uv$ and merge duplicate edges, so the result $G/e$ is again a simple graph. This is not deletion of $uv$, which keeps both original vertices.^[ref]

The left example contracts $uv$ in the cycle graph $C_4$ to give $\{wa,ab,bw\}$. Its endpoints have no common neighbours, so four edges become three. The right example contracts $uv$ in the triangle in a graph with edges $uv,ua,va$. Its endpoints have the single common neighbour $a$, so the two edges $ua,va$ become one edge $wa$: three edges become one.

For any such edge contraction, the exact counts areThe contracted edge accounts for one loss. Each common neighbour $z$ contributes exactly one further loss, since $uz,vz$ become a single edge $wz$. All other edges correspond one-to-one to surviving edges. In particular the new degree of a vertex isThese formulas depend on simplifying to a simple graph; a multigraph convention that retains duplicate edges has different counts. In the six-vertex branch set of a graph minor diagram, contracting $12$ loses one edge, contracting the resulting $34$ loses two because of its common neighbour $A$, and contracting $56$ loses one. Hence its seven edges become the three edges of $K_3$: $7-1-2-1=3$.

An edge contraction can increase a degree of a vertex even while decreasing both total counts. For example, contracting $uv$ in the tree with edges $uv,ua,ub,vc,vd$ merges two vertices of degree of a vertex three into one of degree of a vertex four. This is the mechanism behind the distinction between a graph minor and a topological minor.

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 \(G\).