A Graph Morphism is a structure preserving mapping of nodes and edges. Morphisms can be defined as either Injective where layout is preserved, Non-Injective where layout can change and nodes may merge, and Bijective where an isomorphic change is made to the graph.
A graph rule can be mathematically defined as $r=<L <- K -> R>$ where $L <- K$ is removed, $K -> R$ is added and $K$ is preserved. All rules applied must implicitly comply with the dangling condition, where no edge may be connected to a deleted node. A group of rules can be used to make a Graph Transformation System $<ccL, ccR>$ containing the labels and rules of the system, or described as a Well Structured Flow Diagram.
In addition to the facilities provided by graph rules, connected grammars also allow the use of non-terminal symbols and a start-graph from which all graphs compliant with those rules may be generated. Every graph in a connected grammar must be sound (non-empty and connected) and complete (every non-empty connected graph over $ccL$ is included into the connected grammar). A connected grammar is given as $C=<ccL, ccN, ccR, Start>$ where $ccL$ is the terminal alphabet, $ccN$ is the non-terminal alphabet, $ccR$ is the rules of the grammar, and Start is the start graph.