of adjacent vertices are assigned the same colour. The *chromatic number* $\chi(G)$ is the minimum number $k$ such that $G$ has a $k$-colouring.

A graph $H$ is a *minor* of a graph $G$ if a graph isomorphic to $H$ can be constructed from $G$ by vertex deletion, edge deletion and edge contraction. *Edge contraction* means to take an edge $vw$ and replace $v$ and $w$ with a new vertex $x$ adjacent to all vertices originally adjacent to $v$ or $w$. If $H$ is a minor of $G$, say that $G$ has an $H$-minor.

The *Hadwiger number* ${\textsf{\textup{had}}}(G)$ is the order of the largest complete minor of $G$. The Hadwiger number is most relevant to *Hadwiger’s Conjecture* [@Hadwiger43], often considered one of the most important unsolved conjectures in graph theory, which states that $\chi(G) \leq {\textsf{\textup{had}}}(G)$. Hadwiger’s Conjecture can be seen as an extension of the Four Colour Theorem, since every planar graph has ${\textsf{\textup{had}}}(G) \leq 4$. While the conjecture remains unsolved in general, it has been proved for ${\textsf{\textup{had}}}(G) \leq 5$ [@RST-Comb93].

Given a graph $H$, an *$H$-model* of $G$ is a set of pairwise vertex-disjoint connected subgraphs of $G$, called *branches*, indexed by the vertices of $H$, such that 