Explain this problem with a graph to understand and explain it step by step.
a) Show that if G is a 2-connected graph containing a vertex that is adjacent to at least three vertices of degree 2, then G is not hamiltonian.
b) The subdivision graph S(G) of a graph G is that graph obtained from G by replacing each edge uv of G by a vertex w and edges uw and vw. Determine, with proof, all graphs G for which S(G) is hamiltonian.
Hamiltonian Graphs and 2-Connected Graphs are investigated. The solution is detailed and well presented.