Graphs, contraction mappings and minors

This paper contains a review of topics on morphisms, contractions and minors, and several basic results that have been used repeatedly in the literature. Properties of contraction mappings and of cyclic connectivity are reviewed. The binary relation of minor inclusion is not far from being a partial order in any family of finite graphs. We also prove that if G is a 3-connected, internally 4-connected cubic graph that is not cyclically 4-connected then G ≃ K4 or G ≃ K2□K3. Diagrams as in other mathematical subjects such as algebra ought to be used for graph theory. Diagrams make concepts precise and clear, and avoid long verbal descriptions.