if $vw \in E(H)$, then there exists an edge between the branches indexed by $v$ and $w$. If $G$ has an $H$-model, then repeatedly contract the edges inside each branch and delete extra vertices and edges to obtain a copy of $H$. Thus if $G$ has an $H$-model, then $H$ is a minor of $G$. Similarly, if $H$ is a minor of $G$, “uncontract" each vertex in the minor to obtain an $H$-model of $G$. Models are helpful when dealing with questions relating to minors, as they describe how the $H$-minor “sits” in $G$.

Brambles
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Two subgraphs $A$ and $B$ of a graph $G$ *touch* if $V(A)\cap V(B)\neq\emptyset$, or some edge of $G$ has one endpoint in $A$ and the other endpoint in $B$. A *bramble* in $G$ is a set of connected subgraphs of $G$ that pairwise touch. A set $S$ of vertices in $G$ is a *hitting set* of a bramble ${\ensuremath{\mathcal{B}}}$ if $S$ intersects every element of ${\ensuremath{\mathcal{B}}}$. The *order* of ${\ensuremath{\mathcal{B}}}$ is the minimum size of a hitting set. The *bramble number* of $G$ is the maximum order of a bramble in $G$. Brambles were first defined by @SeymourThomas-JCTB93, where they were called *screens of 