Open Graphs and Monoidal Theories

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph. While the category of open-graphs is not itself adhesive, we introduce the notion of a selective adhesive functor, and show that such a functor embeds the category of open-graphs into the ambient adhesive category of typed graphs. Using this functor, the category of open-graphs inherits “enough adhesivity” from the category of typed graphs to perform double-pushout (DPO) graph rewriting. A salient feature of our theory is that it ensures rewrite systems are “type-safe” in the sense that rewriting respects the inputs and outputs. This formalism lets us safely encode the interesting structure of a computational model, such as evaluation dynamics, with succinct, explicit rewrite rules, while the graphical representation absorbs many of the tedious details. Although topological formalisms exist for string diagrams, our construction is discreet, finitary, and enjoys decidable algorithms for composition and rewriting. We also show how open-graphs can be parametrised by graphical signatures, similar to the monoidal signatures of Joyal and Street, which define types for vertices in the diagrammatic language and constraints on how they can be connected. Using typed open-graphs, we can construct free symmetric monoidal categories, PROPs, and more general monoidal theories. Thus open-graphs give us a handle for mechanised reasoning in monoidal categories.

💡 Research Summary

This paper presents a concrete, finitary construction of string diagrams using a specially typed class of graphs called open‑graphs. In a string diagram, edges may have free ends that serve as inputs or outputs; open‑graphs capture this by augmenting ordinary directed graphs with distinguished vertices called ports. Ports may remain unconnected, thereby representing the external interface of a diagram.

A major technical obstacle is that the category of open‑graphs (OGraph) is not adhesive: not every pair of morphisms admits a pushout, which is essential for the double‑pushout (DPO) approach to graph rewriting. To overcome this, the authors introduce the notion of a selective adhesive functor (F : \text{OGraph} \to \text{TGraph}), where TGraph is the well‑known adhesive category of typed graphs. The functor embeds each open‑graph into a typed graph by interpreting ports as specially labelled vertices. Although the embedding is not full, it is selective: it preserves precisely those pushouts needed for DPO rewriting of open‑graphs. Consequently OGraph inherits “enough adhesivity” from TGraph, allowing the standard DPO machinery (pushout complements, pushout squares, and rule application) to be used safely on open‑graphs.

The DPO framework is then shown to be type‑safe for open‑graphs. When a rewrite rule (L \to R) is applied via a match (M), the ports in the match are required to respect the input‑output arities of the rule. The selective adhesive functor guarantees that the resulting rewritten graph still has the same set of input and output ports, preserving the diagram’s external interface. This property is crucial for modelling physical processes, quantum circuits, or logical gates where the boundary of a system must remain well‑typed throughout transformations.

Beyond the rewriting theory, the paper develops a graphical signature concept, an analogue of Joyal‑Street’s monoidal signatures but formulated for vertices and ports of open‑graphs. A graphical signature specifies, for each vertex type, the number and types of its input and output ports. Open‑graphs that conform to a given signature form a subcategory that is equivalent to the free symmetric monoidal category generated by that signature. Hence, by choosing appropriate signatures one can construct PROPs, Lawvere theories, or more general monoidal theories directly as categories of typed open‑graphs. Composition in these categories is simply the gluing of matching ports, eliminating the need for complex topological constructions.

Algorithmically, the authors provide decidable procedures for three fundamental operations: (1) checking whether a typed open‑graph conforms to a signature, (2) computing the composition of two open‑graphs by identifying compatible ports, and (3) performing DPO rewriting while preserving type safety. Because open‑graphs are finite, discrete structures, all these procedures run in polynomial time with respect to the size of the graphs, making the framework amenable to implementation in automated reasoning tools.

In comparison with existing topological formalisms for string diagrams (e.g., planar isotopy classes, box‑diagrams), the open‑graph approach is discrete, finitary, and algorithmically tractable. It retains the expressive power of string diagrams while providing a solid categorical foundation that supports mechanised reasoning, verification, and transformation of diagrammatic models. The paper thus bridges the gap between high‑level diagrammatic reasoning and low‑level computational manipulation, opening the way for practical applications in quantum computing, circuit design, tensor network optimisation, and any domain where compositional structures are naturally expressed as string diagrams.