# Splitting off

If the degree of s is even, a complete splitting off at s means that we arrange the edges incident to s into pairs $(su_1,sv_1),\dots,(su_k,sv_k)$, and we split off all of these pairs. Thus s becomes isolated and the new edges are $\{u_1v_1,\dots,u_kv_k\}$.
The reverse operation of a complete splitting off is called pinching: if G=(V,E) is an undirected graph and $\{u_1v_1,\dots,u_kv_k\}\subseteq E$, then pinching this set of edges means deleting them from the graph, adding a new node s, and adding edges $su_1,sv_1,\dots,su_k,sv_k$.