I cannot determine if this graph is planar. I have tried all tests but cannot see

Question

I am working with a professor on a research paper and we believe this graph is a counter example to our work, but we need to be sure. We cannot determine if this graph is planar and we need to for our research.

If you are able to see a subdivision of a K5 or a K3,3 as a subgraph of the image, then it is non planar, but we cannot find one, but we also cannot make it planar... please help. https://i.stack.imgur.com/FNPTL.png

Answer 1

The graph is not planar. There is a $K_5$ configuration.

Answer 2

(Not an answer, just a long list of comments.)

Wiki article on planarity testing https://en.wikipedia.org/wiki/Planarity_testing Since this is for research, I would recommend sitting down with an algorithm and implementing it.

Or, I would put it in a program like Sage, it probably has a method to find if the graph is planar. Here was a link I found: http://doc.sagemath.org/html/en/reference/graphs/sage/graphs/schnyder.html whether it has a subdivision (etc. etc.) should be able to be seen by computation.

Answer 3

Merge all the vertices that have not been labelled into one vertex, call it $5$ then $1,2,3,4,5$ will be a graph isomorphic to $K_5$.