Image-Type Labellings/Colorings In Graph Homomorphisms And Graphic Lattices

In topological authentication for encrypting networks wholly we define matchings of dual-type and image-type labellings/colorings, such as totally-dual $(W_{f},\ W_{g})$-type total-image labellings/colorings, uniformly dual-image $(W_{f},\ W_{g})$-type labellings/colorings; and moreover we define graph edge-homomorphism from a graph to another graph, graph edge-homomorphism, dual-image graph base-homomorphism, matching of $(W_{f},\ W_{g})$-type number strings. The Number String Decomposition Problem is proposed for resisting possibly the attack of supercomputers and quantum computing.