If $G=(V,E)$ is a finite, simple, undirected graph, then by $\eta(G)$ we denote the maximum integer $n\in \mathbb{N}$ such that $K\_n$ is a [minor](https://en.wikipedia.org/wiki/Graph_minor) of $G$. If $e\in E$ we write $G\setminus e$ to denote the graph $(V, E \setminus \{e\})$.


Is there a finite graph $G=(V,E)$ and $e\in E$ such that $\eta(G\setminus e) < \eta(G)-1$?



Effect of removing an edge on Hadwiger number
**No,** there is no such graph. Suppose $\eta(G)=n$. Let $T\_1, \dots, T\_n$ be a collection of vertex disjoint trees in $G$ such that for all distinct $i,j \in [n]$, there is an edge $e(ij) \in E(G)$ between $T\_i$ and $T\_j$. Consider an arbitrary edge $e \in E(G)$. If $e=e(ij)$ for some $i,j$, then removing $T\_i$ (or $T\_j$) yields a model of $K\_{n-1}$ in $G \setminus e$. If $e \in E(T\_i)$ for some $i$, then removing $T\_i$ yields a model of $K\_{n-1}$ in $G \setminus e$. Otherwise, $\eta(G \setminus e) = n$. 


