Maximal Digraphs With Respect to Primitive Positive Constructibility

TL;DR

This paper characterizes the structure of finite directed graphs under primitive positive constructibility, identifying a hierarchy with a greatest element, a greatest lower bound, and a complete description of submaximal graphs.

Contribution

It provides a complete description of the greatest lower bounds of the graph with two vertices and one edge, enriching the understanding of the constructibility order.

Findings

$P_1$ is the greatest element in the constructibility order.

$P_2$ is the greatest lower bound of $P_1$.

All graphs not equivalent to $P_1$ or $P_2$ are below a submaximal graph.

Abstract

We study the class of all finite directed graphs up to primitive positive constructability. The resulting order has a unique greatest element, namely the graph $P_1$ with one vertex and no edges. The graph $P_1$ has a unique greatest lower bound, namely the graph $P_2$ with two vertices and one directed edge. Our main result is a complete description of the greatest lower bounds of $P_2$; we call these graphs submaximal. We show that every graph that is not equivalent to $P_1$ and $P_2$ is below one of the submaximal graphs.