Let $ G = \left(V, E\right) $ be a simple path such that $ \left|V\right| = 10 $. Let $ x, y $ be the leaves of $ G $. Two new nodes, $ v, w $, were added to $ V $ such that there is an edge between $ v, w $ and an edge between every $ u \in V $ and $ v, w $. Another vertex $ t $ was added to $ G $ and the following edges too: $tw, tx, ty, tv $. Prove $ G $ is not planar. Is my proof correct?
We shall treat the path $ P = x - v_1 - v_2 \ldots v_8 - y $ as an edge between $ x $ and $ y $. We will now look at the edges we have in the subgraph which only has the vertices $ x, y, t, w, v $.
$ t - \{x, y, v, w\} $
$ x - \{y, v, w, t\} $
$ v - \{y, x, w, t\} $
$ w - \{y, x, v, t\} $
$ y - \{x, v, w, t\} $
Therefore, $ K_5 $ is a subgraph of $ G $, which means $ G $ is not planar.