Biclosed monoidal structures on the categories of digraphs and graphs

TL;DR

This paper demonstrates that the categories of directed and undirected reflexive graphs possess exactly two distinct biclosed monoidal structures, highlighting a fundamental property of these graph categories.

Contribution

It identifies and characterizes the only two biclosed monoidal structures on categories of reflexive graphs, providing a complete classification.

Findings

Categories of directed and undirected reflexive graphs have exactly two biclosed monoidal structures.

The identified structures are unique up to isomorphism.

This classification advances understanding of monoidal structures in graph categories.

Abstract

We show that the categories of directed and undirected reflexive graphs carry exactly two (up to isomorphism) biclosed monoidal structures.