I am supposed to show that the following graph is not hamiltonian.
In the lecture I am attending to we have looked at only one criteria to determine if a graph is not hamiltonian which is that for $\kappa(G-S) \leq |S|$ for any subset $S \subseteq V(G)$ for a hamiltonian graph.
Furthermore I have seen some other criteria of parts of graphs on which one might be able to determine that no hamilton circle can exist however I was not able to apply them to this graph.
Any hints/tips would be greatly appreciated
If you number the nodes from top to bottom and left to right, then $S=\{1,5,6,7,11\}$ certifies that $G$ is not Hamiltonian. Removing $S$ (the red x nodes below) yields $6$ connected components (which happen to have one node each), so $\kappa(G-S) = 6 > 5 = |S|$.