Complement of 7-vertex planar graph

Question

Let $G$ be a 7-vertex planar graph.
Does exist such $G$ that complement of it = $\overline{G}$ that doesn't include 3-vetrex clique?

I've tried for $\overline{G}$ as graph where all vertexes have degree 2 and $\overline{G}$ doesn't include 3-vertex clique, but I can't proof that $G$ is plannar in this case.

Answer 1

It is equivalent to ask for $G$ to be a $7$-vertex planar graph with no $3$-vertex independent set: that is, with independence number $2$.

Searching the House of Graphs database with these criteria brings up 16 different examples. For example, this graph is the union of a $K_4$ and a $K_3$; it's planar, and its complement is the complete bipartite graph $K_{4,3}$, which does not contain a $3$-vertex clique because it has no odd cycles.

To conduct this search, select the following options:

• Require a specific (numeric) value for an invariant: Number of Vertices equal to 7
• Require a specific (numeric) value for an invariant: Independence Number less than 3
• Only consider certain classes of graphs: Planar