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Given a graph G how does one check if it can be decomposed into 2 planar graphs other than enumeration.

Note: my given graph has 52 edges and 11 nodes.

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Quick edit: the graph has an edge from 4 to 1*

RAH
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1 Answers1

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Lemma:

The thickness of $K_n$ is $\lfloor\frac{n+7}{6}\rfloor$ when $n\neq 9,10$ and for cases $n=9,10$ it is $3$.

So if there exists a $K_a$ with $a\geq 9$ in your graph (lets call it $G$), that subgraph has a thickness of at least $3$, so $G$ has a thickness of al least $3$

So for your graph to have a thickness of $2$, it must not have any $K_9$s as subgraphs. Using Turan's theorem, the maximum number of edges of a graph with $11$ vertices which is $K_9-$free is $52$ and if this bound is touched by some graph, it is the Turan graph $T(11,8)$.

Thus, your graph does not have a thickness of $2$ if it is not the Turan graph. If your graph is the Turan graph $T(11,8)$, then observe that it doesnt actually have $52$ edges so again, there is a $K_9$, so your graph has a thickness of at least $3$.

So you graph has a thickness $\geq 3$.

Some closing remarks:

It is NP-hard to compute the thickness of a given graph, and NP-complete to test whether the thickness is at most two.

It is pretty hard to just test the thickness of a graph, as cited above from Wikipedia, especially if you do not have the graph and only the number of vertices and edges are given.

  • Do you have a reference for the lemma? – Servaes Oct 27 '20 at 13:51
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    Hello there. There are actually many more lemmas reagrding thickness. See this: https://domino.mpi-inf.mpg.de/internet/reports.nsf/efc044f1568a0058c125642e0064c817/9097519627562713c125630a0047835e/$FILE/MPI-I-96-1-009.pdf –  Oct 27 '20 at 13:54
  • Why are you saying that the Turán graph doesn't actually have $52$ edges? Start with $K_{11}$, which has $55$ edges. Then delete $3$ independent edges $v_1w_1, v_2w_2, v_3w_3$ to get $52$. A clique in the resulting graph can only have one of ${v_1, w_1}$, only one of ${v_2, w_2}$, and only one of ${v_3, w_3}$, so it can have at most $11-3=8$ edges, and we have a $52$-edge graph which is $K_9$-free. – Misha Lavrov Oct 30 '20 at 21:55