# An outerplanar G graph such that $G+k_1$ is not planar graph

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?

• What is definition of $k1$? – coffeemath Nov 22 '18 at 8:08
• Isn't $G+k_1$ planar when $G$ is outerplanar, precisely because of the sentence "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."? – Inactive - avoiding CoC Nov 22 '18 at 11:48
• I agree with that, iff $k_1$ is really supposed to mean the regular graph on one vertex (aka graph with a single vertex and no edges) – Ingix Nov 22 '18 at 12:10
• A graph is outerplanar if and only if it does not contain $K_4$ or $K_{2,3}$ as minor. – mathpadawan Nov 24 '18 at 11:15