The Kuratowski's theorem says, that a graph is planar if, and only if it doesn't contain a subgraph that is a subdivision of [math]K_5[/math] or [math]K_{3,3}[/math]. We are now using instead the more general theorem of Klaus Wagner and look for minors of [math]K_5[/math] and [math]K_{3,3}[/math]. On the application below you can see the Petersen graph.[br]Try to find the [math]K_5[/math] and [math]K_{3,3}[/math] minor in the Petersen graph.[br]-You can merge vertices that are connected by edges by moving them together and left-clicking them.[br]-You can use the rubber-tool to delete vertices and edges.[br]-To restore the original version press the refresh-button on the right top.[br]-You can rearrange the vertices of the [math]K_5[/math] and [math]K_{3,3}[/math] and try to overlap them with the Petersen graph.

If you have problems merging two vertices, try to wait a bit after clicking a vertex. Also check if the vertices are connected through an edge. You can also try to switch to the html 5 version.[br]You will have to reset the application to solve the [math]K_5[/math] problem. Therefore just reload the page.