What is a graph that contains subdivision but not minors and vice versa?

Question

I am looking for graphs which satisfy the following conditions. (I have tried finding some possible solutions but no way to confirm them)

(a). a K5 as a minor but no K5-subdivision? ans. Petersen Graph?

(b) a K3,3 as minor but no K3,3-subdivision? ans. Toroidal Graph?

(c) a K5-subdivision but no K5 as minor?

(d) a K3,3-subdivision but no K3,3 as minor?

Answer 1

It should be straight forward to prove the following statement:

Let $G$ be a graph. All induced subgraphs of $G$ are subgraphs of $G$. All subgraphs of $G$ are topological minors of $G$. All topological minors of $G$ are minors of $G$.

The other direction is not true. As you mention in your post, there exists examples of minors which are not topological minors (e.g. the Peterson graph with minor $K_5$).

To see why this is true, remember the definitions of what it means to be each:

A graph $H$ is an induced subgraph of a graph $G$ if there is a sequence of vertex removals that transforms $G$ into $H$.

A graph $H$ is a subgraph of a graph $G$ if there is a sequence of vertex removals and/or edge removals that transforms $G$ into $H$.

A graph $H$ is a topological minor of a graph $G$ if there is a sequence of vertex removals, edge removals, and/or suppression of vertices of degree 2 that transforms $G$ into $H$.

A graph $H$ is a minor of a graph $G$ if there is a sequence of vertex removals, edge removals, suppression of vertices of degree 2, and/or edge contractions that transforms $G$ into $H$.

These sequences are allowed to be of length zero, implying that a graph is always an induced subgraph, subgraph, topological minor, and minor of itself.

If you know something is, for example, a subgraph of $G$, then there is some sequence of transformations from a list of allowable transformations. Since each of those transformations are allowable for, say, showing it is a minor, the same sequence of transformations can be used to show that it is a minor.