Definitions
An edge contraction is an operation which removes an edge from a graph while simultaneously merging the two vertices it used to connect. An undirected graph H is a minor of another undirected graph G if a graph isomorphic to H can be obtained from G by contracting some edges, deleting some edges, and deleting some isolated vertices. The order in which a sequence of such contractions and deletions is performed on G does not affect the resulting graph H.
Graph minors are often studied in the more general context of matroid minors. In this context, it is common to assume that all graphs are connected, with self-loops and multiple edges allowed (that is, they are multigraphs rather than simple graphs); the contraction of a loop and the deletion of a cut-edge are forbidden operations. This point of view has the advantage that edge deletions leave the rank of a graph unchanged, and edge contractions always reduce the rank by one.
In other contexts (such as with the study of pseudoforests) it makes more sense to allow the deletion of a cut-edge, and to allow disconnected graphs, but to forbid multigraphs. In this variation of graph minor theory, a graph is always simplified after any edge contraction to eliminate its self-loops and multiple edges.
Read more about this topic: Graph Minor
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