Constructing graphs with limited resources

TL;DR

This paper explores the minimal physical resources needed to construct various graphs sequentially, revealing that simple graphs like threshold graphs require minimal instructions and no memory, while more complex graphs need additional resources.

Contribution

It introduces a resource-based framework for graph construction, identifying the minimal instructions and memory needed for different graph classes and establishing connections to random graph models.

Findings

Threshold graphs require only one bit of instruction per vertex.

With one bit of instruction and memory, perfect graphs beyond threshold graphs are constructible.

Constructing any graph with random bits alone matches the complexity of Erdős-Rényi random graphs.

Abstract

We discuss the amount of physical resources required to construct a given graph, where vertices are added sequentially. We naturally identify information -- distinct into instructions and memory -- and randomness as resources. Not surprisingly, we show that, in this framework, threshold graphs are the simplest possible graphs, since the construction of threshold graphs requires a single bit of instructions for each vertex and no use of memory. Large instructions without memory do not bring any advantage. With one bit of instructions and one bit of memory for each vertex, we can construct a family of perfect graphs that strictly includes threshold graphs. We consider the case in which memory lasts for a single time step, and show that as well as the standard threshold graphs, linear forests are also producible. We show further that the number of random bits (with no memory or instructions) needed to construct any graph is asymptotically the same as required for the Erd\H{o}s-R\'enyi random graph. We also briefly consider constructing trees in this scheme. The problem of defining a hierarchy of graphs in the proposed framework is fully open.