## A definition of minor (in graph theory)

Many people I talk to about graph theory feel some uneasiness when it comes to the notion of “minor”. I want to try to alleviate this feeling by providing the definiton of minor that I work with.

First an easy definition. If $G$ is a simple, undirected graph and $S, T\subseteq V(G)$ are non-empty and disjoint, we say that $S, T$ are connected to each other if there are $s\in S, t\in T$ such that $\{s,t\}\in E(G)$.

Let $G, H$ be simple, undirected graphs. We say that $G$ is a minor of $H$ if there is a collection ${\cal S}$ of non-empty, mutually disjoint, and connected subsets of $V(H)$ and a bijection $\varphi:V(G) \to {\cal S}$ such that

whenever $v,w\in V(G)$ and $\{v,w\}\in E(G)$ then the sets $\varphi(v)$ and $\varphi(w)$ are connected to each other in $H$.