Graph Theory proving planarity I have these set's of graphs: 

I used Euler's inequality, and the 4 color theorem which resulted in a inconclusive result. Using Kuratowski's theorem I was unable to create a K3 3 or K5 graph. So does this prove that the graphs are planar? How do we really know that there does not exist a K3 3 or K5? I did use planar embedding and was able to put it in a form where the edges do not overlap, but the method does not seem rigorous enough to prove planarity. 

In the first graph, there is a subdivision or minor of $K_5$ formed by $a$, $\{b,c\}$, $d$, $\{e,f\}$, $g$.
Therefore it is not planar.
The second graph has a planar embedding. This is the definition of a graph being planar, so I have no idea what you mean by whether it is not rigorous.
