Editing Graph minor (section)

===Immersion minor===
A graph operation called ''lifting'' is central in a concept called ''immersions''.  The ''lifting'' is an operation on adjacent edges.  Given three vertices ''v'', ''u'', and ''w'', where ''(v,u)'' and ''(u,w)'' are edges in the graph, the lifting of ''vuw'', or equivalent of ''(v,u), (u,w)'' is the operation that deletes the two edges ''(v,u)'' and ''(u,w)'' and adds the edge ''(v,w)''.  In the case where ''(v,w)'' already was present, ''v'' and ''w'' will now be connected by more than one edge, and hence this operation is intrinsically a multi-graph operation.

In the case where a graph ''H'' can be obtained from a graph ''G'' by a sequence of lifting operations (on ''G'') and then finding an isomorphic subgraph, we say that ''H'' is an immersion minor of ''G''.
There is yet another way of defining immersion minors, which is equivalent to the lifting operation.  We say that ''H'' is an immersion minor of ''G'' if there exists an injective mapping from vertices in ''H'' to vertices in ''G'' where the images of adjacent elements of ''H'' are connected in ''G'' by edge-disjoint paths.

The immersion minor relation is a well-quasi-ordering on the set of finite graphs and hence the result of Robertson and Seymour applies to immersion minors.  This furthermore means that every immersion minor-closed family is characterized by a finite family of forbidden immersion minors.

In [[graph drawing]], immersion minors arise as the [[planarization]]s of [[planar graph|non-planar graphs]]: from a drawing of a graph in the plane, with crossings, one can form an immersion minor by replacing each crossing point by a new vertex, and in the process also subdividing each crossed edge into a path. This allows drawing methods for planar graphs to be extended to non-planar graphs.{{sfnp|Buchheim|Chimani|Gutwenger|Jünger|2014}}