In computer science, graph transformation, or graph rewriting, concerns the technique of creating a new graph out of an original graph algorithmically. It has numerous applications, ranging from software engineering (software construction and also software verification) to layout algorithms and picture generation. Graph transformations can be used as a computation abstraction. The basic idea is that if the state of a computation can be represented as a graph, further steps in that computation can then be represented as transformation rules on that graph. Such rules consist of an original graph, which is to be matched to a subgraph in the complete state, and a replacing graph, which will replace the matched subgraph. Formally, a graph rewriting system usually consists of a set of graph rewrite rules of the form L → R {\displaystyle L\rightarrow R} , with L {\displaystyle L} being called pattern graph (or left-hand side) and R {\displaystyle R} being called replacement graph (or right-hand side of the rule). A graph rewrite rule is applied to the host graph by searching for an occurrence of the pattern graph (pattern matching, thus solving the subgraph isomorphism problem) and by replacing the found occurrence by an instance of the replacement graph. Rewrite rules can be further regulated in the case of labeled graphs, such as in string-regulated graph grammars. Sometimes graph grammar is used as a synonym for graph rewriting system, especially in the context of formal languages; the different wording is used to emphasize the goal of constructions, like the enumeration of all graphs from some starting graph, i.e. the generation of a graph language – instead of simply transforming a given state (host graph) into a new state.
Graph rewriting approaches
Algebraic approach The algebraic approach to graph rewriting is based upon category theory. The algebraic approach is further divided into sub-approaches, the most common of which are the double-pushout (DPO) approach and the single-pushout (SPO) approach. Other sub-approaches include the sesqui-pushout and the pullback approach. From the perspective of the DPO approach a graph rewriting rule is a pair of morphisms in the category of graphs and graph homomorphisms between them: r = ( L ← K → R ) {\displaystyle r=(L\leftarrow K\rightarrow R)} , also written L ⊇ K ⊆ R {\displaystyle L\supseteq K\subseteq R} , where K → L {\displaystyle K\rightarrow L} is injective. The graph K is called invariant or sometimes the gluing graph. A rewriting step or application of a rule r to a host graph G is defined by two pushout diagrams both originating in the same morphism k : K → D {\displaystyle k\colon K\rightarrow D} , where D is a context graph (this is where the name double-pushout comes from). Another graph morphism m : L → G {\displaystyle m\colon L\rightarrow G} models an occurrence of L in G and is called a match. Practical understanding of this is that L {\displaystyle L} is a subgraph that is matched from G {\displaystyle G} (see subgraph isomorphism problem), and after a match is found, L {\displaystyle L} is replaced with R {\displaystyle R} in host graph G {\displaystyle G} where K {\displaystyle K} serves as an interface, containing the nodes and edges which are preserved when applying the rule. The graph K {\displaystyle K} is needed to attach the pattern being matched to its context: if it is empty, the match can only designate a whole connected component of the graph G {\displaystyle G} . In contrast a graph rewriting rule of the SPO approach is a single morphism in the category of labeled multigraphs and partial mappings that preserve the multigraph structure: r : L → R {\displaystyle r\colon L\rightarrow R} . Thus a rewriting step is defined by a single pushout diagram. Practical understanding of this is similar to the DPO approach. The difference is, that there is no interface between the host graph G and the graph G' being the result of the rewriting step. From the practical perspective, the key distinction between DPO and SPO is how they deal with the deletion of nodes with adjacent edges, in particular, how they avoid that such deletions may leave behind "dangling edges". The DPO approach only deletes a node when the rule specifies the deletion of all adjacent edges as well (this dangling condition can be checked for a given match), whereas the SPO approach simply disposes the adjacent edges, without requiring an explicit specification. There is also another algebraic-like approach to graph rewriting, based mainly on Boolean algebra and an algebra of matrices, called matrix graph grammars.
Determinate graph rewriting Yet another approach to graph rewriting, known as determinate graph rewriting, came out of logic and database theory. In this approach, graphs are treated as database instances, and rewriting operations as a mechanism for defining queries and views; therefore, all rewriting is required to yield unique results (up to isomorphism), and this is achieved by applying any rewriting rule concurrently throughout the graph, wherever it applies, in such a way that the result is indeed uniquely defined.
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