Structured Decompositions: Structural and Algorithmic Compositionality

TL;DR

Structured Decompositions develops a category-theoretic framework that unifies width measures across graphs, groups, hypergraphs, and dynamical systems by modeling decompositions as colimits of diagrams and organizing them into width categories via $\oldmath{$\mathbf{w}$}_{\Gamma}$. It introduces sd-categories, width categories, and sd-functors, proving key properties like width monotonicity and the equivalence of chordal width with width in complete width categories, and connects structured decompositions to chordal completions. A broad set of examples shows classical invariants such as treewidth, pathwidth, layered treewidth, and hypergraph treewidth arise as instances of $\mathbf{w}_{\Gamma}$, while Bass-Serre theory and hybrid dynamical systems illustrate the framework’s cross-domain reach. The work lays groundwork for compositional, width-aware algorithms and a unifying perspective on structural complexity across mathematics and applied domains.

Abstract

We introduce structured decompositions, category-theoretic structures which simultaneously generalize notions from graph theory (including treewidth, layered treewidth, co-treewidth, graph decomposition width, tree independence number, hypergraph treewidth and H-treewidth), geometric group theory (specifically Bass-Serre theory), and dynamical systems (e.g. hybrid dynamical systems). We define width functors, which provide a compositional way to analyze and relate different structural complexity measures, and establish a general duality between decompositions and completions of objects.

Figures (2)

• Figure 1: The first four nonempty complete graphs.
• Figure 2: An example of a tree decomposition of a graph $G$.

Theorems & Definitions (226)

• Definition 2.1.1
• Definition 2.1.2
• Definition 2.2.1
• Example 2.2.2
• Definition 2.3.1
• Example 2.3.2
• Definition 2.3.3
• Example 2.3.4
• Definition 2.4.1
• Remark 2.4.2