The diagram shows a graph \(\mathbf { G }\) with 9 vertices and 9 edges.
\includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-08_188_204_411_708}
\includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-08_184_204_415_1105}
\includegraphics[max width=\textwidth, alt={}, center]{e47eb41e-0a4b-4865-a8ff-6c9978495ee0-08_183_204_612_909}
State the minimum number of edges that need to be added to \(\mathbf { G }\) to make a connected graph. Draw an example of such a graph.
State the minimum number of edges that need to be added to \(\mathbf { G }\) to make the graph Hamiltonian. Draw an example of such a graph.
State the minimum number of edges that need to be added to \(\mathbf { G }\) to make the graph Eulerian. Draw an example of such a graph.
A complete graph has \(n\) vertices and is Eulerian.
State the condition that \(n\) must satisfy.
In addition, the number of edges in a Hamiltonian cycle for the graph is the same as the number of edges in an Eulerian trail. State the value of \(n\).