If $G=(V,E)$ is a finite, simple, undirected graph, the Hadwiger number $\eta(G)$ is defined to be the size of the largest complete minor of $G$.

Is there a finite graph $G$ with the following property?

Whenever vertices of distance $2$ are collapsed, the Hadwiger number increases.

If $G=(V,E)$ is a finite, simple, undirected graph, the Hadwiger number $\eta(G)$ is defined to be the size of the largest complete minor of $G$.

Is there a finite graph $G=(V,E)$ with the following property?

Whenever $v,w\in V$ have distance $2$, the Hadwiger number of the graph obtained by collapsing $v,w$ is strictly larger than $\eta(G)$.

If $G=(V,E)$ is a finite, simple, undirected graph, the Hadwiger number $\eta(G)$ is defined to be the size of the largest complete minor of $G$.

Is there a finite graph $G$ such that whenever vertices of distance $2$ are collapsed, the Hadwiger number increases, but there are non-adjacent points of distance greater than $2$ such that collapsing them leaves the Hadwiger number unchanged?

If $G=(V,E)$ is a finite, simple, undirected graph, the Hadwiger number $\eta(G)$ is defined to be the size of the largest complete minor of $G$.

Is there a finite graph $G$ with the following property?

Whenever vertices of distance $2$ are collapsed, the Hadwiger number increases.