An outerplanar graph is an undirected graph that can be drawn in the plane without crossings in such a way that all of the vertices belong to the unbounded face of the drawing. That is, no vertex is totally surrounded by edges. Alternatively, a graph G is outerplanar if the graph formed from G by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph.
The join $G=G_1+G_2$ of graphs $G_1$ and $G_2$ with disjoint point sets $V_1$ and $V_2$ and edge sets $X_1$ and $X_2$ is the graph union $G_1$ union $G_2$ together with all the edges joining $V_1$ and $V_2$.
Does it exist An outerplanar $G$ graph such that $G+ k_1$ is not planar graph? where $k_1$ is a vertex. can you help me?