Closed symmetric monoidal structures on the category of graphs

TL;DR

This paper characterizes the category of graphs as having exactly two closed symmetric monoidal structures: the box product and the categorical product, clarifying their unique roles.

Contribution

It identifies and proves the existence of precisely two such monoidal structures on the category of graphs, providing a clear structural understanding.

Findings

The category of graphs has exactly two closed symmetric monoidal products.

The two monoidal structures are the box product and the categorical product.

These structures are unique in the category of graphs.

Abstract

We show that the category of (reflexive) graphs and graph maps carries exactly two closed symmetric monoidal products: the box product and the categorical product.