Monoidal Width: Capturing Rank Width

TL;DR

This paper explores monoidal width as a measure of complexity in different categories, demonstrating its ability to capture graph complexity measures like rank width through algebraic and compositional methods.

Contribution

The paper extends the concept of monoidal width to matrix and open graph categories, linking it to the graph complexity measure known as rank width.

Findings

Monoidal width captures tree, path, and branch widths in cospan categories.

In matrix and open graph categories, monoidal width corresponds to rank width.

The approach unifies algebraic and graph-theoretic complexity measures.

Abstract

Monoidal width was recently introduced by the authors as a measure of the complexity of decomposing morphisms in monoidal categories. We have shown that in a monoidal category of cospans of graphs, monoidal width and its variants can be used to capture tree width, path width and branch width. In this paper we study monoidal width in a category of matrices, and in an extension to a different monoidal category of open graphs, where the connectivity information is handled with matrix algebra and graphs are composed along edges instead of vertices. We show that here monoidal width captures rank width: a measure of graph complexity that has received much attention in recent years.