# Effect of removing an edge on Hadwiger number

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 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 $$\eta(G\setminus e) < \eta(G)-1$$?

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, $$\eta(G \setminus e) = n$$.