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(Solved): For \( G \) a graph, let \[ \bar{G}:=\left(V(G),\left(\begin{array}{c} V(G) \\ 2 \end{array}\right) ...
![For \( G \) a graph, let
\[
\bar{G}:=\left(V(G),\left(\begin{array}{c}
V(G) \\
2
\end{array}\right) \backslash E(G)\right)
\]](https://media.cheggcdn.com/study/c0e/c0e00e64-fb2c-4941-a88b-8d8c47e06f37/image)
For \( G \) a graph, let \[ \bar{G}:=\left(V(G),\left(\begin{array}{c} V(G) \\ 2 \end{array}\right) \backslash E(G)\right) \] denote its complement. Show that if \( v(G) \geq 11 \) then at most one of \( G, \bar{G} \) is planar. (Hint: What can you say about the number of edges of \( G \) and \( \bar{G} ? \) ?)
Expert Answer
If G is a graph with n vertices and e edges, then the complement of G, denoted G', has n vertices and n(n-1)/2 - e edges. This is because the compleme