minor relation and topological minor relations are partial orderings of on the class of finite graphs

Question

I have a confusion with proposition 1.73 of the Diestal book on Graph Theory which states that

The minor relation and the topological-minor relation are partial orderings on the class of finite graphs, i.e. they are reflexive, antisymmetric and transitive

I didn't get the idea how to define partial orderings on minor relations

Any idea on how to prove the proposition?

Answer 1

It's not a "partial ordering on minor relations".

The idea is to use the minor relation, that is, the relation defined by $$ (G_1,G_2)\in R \iff G_1 \text{ is a minor of }G_2 $$ as a partial ordering on graphs. In order to prove this relation is a partial ordering, you must prove:

Reflexivity: $G$ is a minor of $G$ itself (for any $G$).

Antisymmetry: If $G$ is a minor of $H$ and $H$ is a minor of $G$, then they're the same graph. (Here you probably need to interpret "are the same graph" as "are isomorphic", or to declare that "a graph" really means an isomorphism class of graphs).

Transitivity: If $G$ is a minor of $H$ and $H$ is a minor of $K$, then $G$ is a minor of $K$.

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Then do all this again for topological minors rather than just minors.