Treewidth via Spined Categories (extended abstract)

TL;DR

This paper introduces a categorical framework called spined categories to define a functorial analogue of treewidth, unifying various treewidth-like invariants across different combinatorial objects.

Contribution

It develops the concept of spined categories and the triangulation functor, providing a unified, categorial approach to treewidth and its variants for graphs and hypergraphs.

Findings

Defines spined categories with extra structure for functorial treewidth analogues

Introduces the triangulation functor as a categorical generalization of treewidth

Recovers classical treewidth and hypergraph treewidth as special cases

Abstract

Treewidth is a well-known graph invariant with multiple interesting applications in combinatorics. On the practical side, many NP-complete problems are polynomial-time (sometimes even linear-time) solvable on graphs of bounded treewidth. On the theoretical side, treewidth played an essential role in the proof of the celebrated Robertson-Seymour graph minor theorem. While defining treewidth-like invariants on graphs and treewidth analogues on other sorts of combinatorial objects (incl. hypergraphs, digraphs) has been a fruitful avenue of research, a direct, categorial description capturing multiple treewidth-like invariants is yet to emerge. Here we report on our recent work on spined categories (arXiv:2104.01841): categories equipped with extra structure that permits the definition of a functorial analogue of treewidth, the triangulation functor. The usual notion of treewidth is recovered as a special case, the triangulation functor of a spined category with graphs as objects and graph monomorphisms as arrows. The usual notion of treewidth for hypergraphs arises as the triangulation functor of a similar category of hypergraphs.