I want to prove the following proposition
$\textbf{proposition}$: $G$ is an $MX$ if and only if $X$ can be obtained from $G$ by a series of edge contractions, i.e if and only if there are graphs $G_{0},...,G_{n}$ and edges $e_{I} \in G_{I}$ such that $G_{0}=G, G_{n} \simeq X$ and $G_{I+1}=G_{I}/ e_{I}$ for all $I<n$.
The hint says use induction on $|G|-|X|$
So first of all I give the definition of a minor that they use
Given a graph $G$ and another graph $X$ and $\{V_{x}|x \in V(X)\}$ is a partition of $V$ into connected subsets such that, for any two vertices $x,y \in X$ there is a $V_{x}-V_{y}$ edge in $G$ if and only if $xy \in E(X)$, we call $G$ an $MX$ and write $G=MX$.
So I'll split my proof into two parts
(1) Given a series of edge contractions in the graph $G$, I need to prove that the graph $X$ obtained is indeed a minor.
So using induction on $|G|-|X|$, suppose $|G|-|X|=1$ in which case I contract a single edge $\{u,v\} \in G$, then clearly the graph $X$ obtained is a minor. Suppose now that the result holds whenever $|G|-|X|<j$. Consider a series of edge contractions $e_{1},...,e_{j}$ we first contract edge $e_{1}=\{u,v\}$ to obtain a graph $G'$ with $|G'|=|G|-1$ where $G'=G/e_{1}$. Now since $|G'|-|X|<j$ it follows that with the edge contractions $e_{2},...,e_{j}$ that $X$ is a minor of $G'$. Then it follows that $X$ is also a minor of $G$ because if the vertex $v_{uv} \in G'$ obtained by contracting $u$ and $v$ in $G$ is in some $V_{x}$ then I can simply replace $v_{uv} \in V_{x}$ with $u$ and $v$ and it follows that $X$ is a minor of $G$.
Now for the second part I need to show
(2) If $X$ is a minor of $G$ that is $G=MX$ then I can indeed obtain $X$ through a sequence of edge contractions.
Again I use induction on $|G|-|X|$. If $|G|-|X|=1$ then $X$ is a minor obtained by contracting a single edge. Suppose now that the result holds for whenever $|G|-|X|<j$ and let $|G|-|X|=j$. Consider a set $V_{x}$ with $|V_{x}|\geq 2$. Pick an edge $e_{0}$ in the connected subset $|V_{x}|$ and contract it to obtain the graph $G'$ where $|G'|=|G|-1$. Then it follows that there exists a sequence of edge contractions $e_{1},...,e_{j-1}$ which allows us to obtain $X$ from $G'$ then along with the edge $e_{0}$, it follows we can obtain $X$ from $G$ by contracting the edges $e_{0},e_{1},...,e_{j-1}$.
Is this proof correct? Are there any details I should add? I appreciate any advice thanks!!
Also I think I'm using the relation that if $G=MX$ and $X=MY$ then $G=MY$ does this require proof itself?