Universal slices of the category of graphs

TL;DR

This paper characterizes the slices of the category of graphs that are algebraically universal, showing that such universality depends on the presence of specific subgraphs within the slicing graph.

Contribution

It provides a complete characterization of algebraically universal slices of the category of graphs based on subgraph containment.

Findings

Algebraic universality occurs iff the slicing graph contains one of four fixed subgraphs.

The paper identifies the exact subgraph structures necessary for universality.

Provides a structural criterion for universality in graph categories.

Abstract

We characterise the slices of the category of graphs that are algebraically universal in terms of the structure of the slicing graph. In particular, we show that algebraic universality is obtained if, and only if, the slicing graph contains one of four fixed graphs as a subgraph.