Decomposing graphs into simpler graphs is one of the central concerns of graph theory. Investigations have revealed deep concepts such as modular decomposition, tree width or rank width, which measure-in different ways-the structural complexity of a graph's topology. Courcelle and others have shown that such concepts can be used to obtain efficient algorithms for families of graphs that are amenable to decomposition (e.g. those that have bounded tree-width). These algorithms, in turn, are of course of use in computer science, where graphs are ubiquitous. In this paper we take the first steps towards understanding notions of decomposition in graph theory compositionally, and more generally, in a categorical setting: category theory, after all, is the mathematics of compositionality. We introduce the concept of ∪−matrices (cup-matrices). Like ordinary matrices, ∪-matrices are the arrows of a PROP: we give a presentation, extending the work of Lafont, and Bonchi, Zanasi and the second author. A variant of ∪-matrices is then used in the development of a novel algebra of simple graphs, the lingua franca of graph theory. The algebra is that of a certain symmetric monoidal theory: ∪-matrices-akin to adjacency matrices-encode the graphs' topology.
Given a classical algebraic structure-e.g. a monoid or groupwith carrier set X, and given a positive integer n, there is a canonical way of obtaining the same structure on carrier set X n by defining the required operations "pointwise". For resource-sensitive algebra (i.e. based on mere symmetric monoidal, not cartesian structure), similar "pointwise" operations are usually defined as a kind of syntactic sugar: for example, given a comonoid structure on X, one obtains a comultiplication on X ⊗ X by tensoring two comultiplications and composing with an appropriate permutation. This is a specific example of a general construction that we identify and refer to as multiplexing. We obtain a general theorem that guarantees that any equation that holds in the base case will hold also for the multiplexed operations, thus generalising the "pointwise" definitions of classical universal algebra.
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