# Splitting off

## Undirected graphs

Let *G=(V+s,E)* be an undirected graph where the degree of *s* is at least two. If *u* and *v* are neighbours of *s*, **splitting off** *su* and *sv* means deleting the two edges *su* and *sv*, and adding an edge *uv* to the graph. Note that if there are several parallel edges between *s* and *u*, then we delete only one of them, except when *u=v*.

If the degree of *s* is even, a **complete splitting off** at *s* means that we arrange the edges incident to *s* into pairs [math](su_1,sv_1),\dots,(su_k,sv_k)[/math], and we split off all of these pairs. Thus *s* becomes isolated and the new edges are [math]\{u_1v_1,\dots,u_kv_k\}[/math].

The reverse operation of a complete splitting off is called **pinching**: if *G=(V,E)* is an undirected graph and [math]\{u_1v_1,\dots,u_kv_k\}\subseteq E[/math], then pinching this set of edges means deleting them from the graph, adding a new node *s*, and adding edges [math]su_1,sv_1,\dots,su_k,sv_k[/math].

## Directed graphs

Splitting off in directed graphs can be defined similarly. Let *D=(V+s,A)* be a directed graph. If *us* and *sv* are in *A*, **splitting off** *us* and *sv* means deleting these two arcs and adding an arc *uv* to the digraph. If the in-degree of *s* equals its out-degree, a **complete splitting off** at *s* is a sequence of splitting of operations at *s* after which *s* becomes isolated. The reverse operation is called **pinching** in digraphs too.