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I am looking for general advice on how to find graph minors (for example $K_{3,3}$ or $K_{5}$) efficiently. I often struggle applying the Kuratowski-Wagner theorem for non-planarity, but want to improve for my upcoming exam.

As for how I currently go about determining planarity, I first spend a few minutes trying to redraw the graphs in a planar way. I move around the vertices that seems to be complicating things (for example in question A below, I might move the two vertices in the square to the outside), but doing so with pen and paper could be quite tedious, and I don't get so many tries under strict time conditions.

If that fails, I move on to finding $K_{3,3}$ or $K_5$ minors. I check if any vertex has degree less than $4$ because if there is, I only need to look for $K_{3,3}$ minors. This actually happens to be the case for all 4 example problems below. However, from here I don't really have a strategy other than just aimlessly applying edge contractions, edge deletions, and vertex deletions. I know that in each step I can't create a vertex of degree less than $2$, but other than that I feel like I just rely on luck.

What are some tips or things you guys keep in mind when trying to find graph minors that could help me out?

Any advice is welcome. Thank you.

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bob
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    Just as an observation, it's not true that if $\delta(G)<4$ then you only need to look for minors of $K_5$, you need there to be no vertices of degree $\geq4$ to arrive at that conclusion, i.e. $\Delta(G)<4$ – Bruno Andrades Apr 08 '25 at 02:24
  • @BrunoAndrades. Thanks for the comment. Could you calify your statement a litte? So if the max degree of the vertices is $<4$, then I need to look for minors of $K_5$?? – bob Apr 14 '25 at 07:09
  • I guess, Bruno means that you do NOT need to look for $K_5$ minors if max degree less than 4... I guess you mean that you do NOT need to look for $K_5$ minors for a specific vertex if that vertex has degree less than 4... – Michael T Apr 22 '25 at 14:27
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    Minors can be formed by contracting edges, which merges vertices together. So that can increase the vertex degrees. This means that even a graph with only degree 3 vertices can have a K5 minor. – Jaap Scherphuis Apr 22 '25 at 15:01
  • @JaapScherphuis That is correct, but still compatible with the claim that you don't need to look for $K_5$ minors in such cases. A non-planar graph with fewer than 5 vertices of degree >3 cannot have a $K_5$ subdivision, so by Kuratowski's theorem it has a $K_{3,3}$ subdivision, which is also a $K_{3,3}$ minor. It might, in addition, have a $K_5$ minor. – Misha Lavrov Apr 22 '25 at 15:50

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It is trial and error (Wikipedia says 'In practice, it is difficult to use Kuratowski's criterion to quickly decide whether a given graph is planar.') and I agree with your strategy:

I find a) difficult to see; it is planar and https://houseofgraphs.org/graphs/748. It can be 'seen' by folding one of the inner vertices outside the square and then pull the opposite corner of the square in... in the right way.

I find it rather easy to find $K_{3,3}$ in b): the left vertex links to only 3, the 2 other vertices link to the same 3. See the comment to find a $K_5$.

For c) you start from one of the 2 vertices with 3 edges and then identify 2 more vertices that also link to these 3 to find 'your' $K_{3,3}$. See the comment to find a $K_5$.

I find d) rather easy to see that you can fold the left vertex into the next triangle and pull the 'diagonal' from top left to bottom right 'out of the diamond'. Hope that is correct...

References

Michael T
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    I would do (b) and (c) by finding K5. In both cases there is an obvious K4 (the vertices arranged in a rectangle), and some of the other vertices that are connected to each other can be merged together by contracting edges, thereby forming a vertex that extends the K4 to a K5. Edit: Actually, with (c) you don't need to merge vertices. The bottom vertex by itself can be the fifth one. – Jaap Scherphuis Apr 22 '25 at 14:56
  • Agree; thanks for pointing that out! – Michael T Apr 22 '25 at 15:01
  • Will this work on a): https://math.stackexchange.com/a/1241614/1511451? – Michael T Apr 22 '25 at 15:18