The Complexity of Growing a Graph

TL;DR

This paper investigates the complexity of designing graph growth processes that start from a single vertex and evolve through vertex and edge operations, aiming to minimize growth steps and excess edges.

Contribution

It introduces the problem of creating growth schedules for target graphs, providing both positive and negative results for schedules with minimal steps and zero excess edges.

Findings

Schedules with sub-linear number of slots are sometimes possible.

Zero excess edges schedules are often computationally hard to find.

The paper characterizes the complexity of various growth schedule scenarios.

Abstract

We study a new algorithmic process of graph growth which starts from a single initial vertex and operates in discrete time-steps, called \emph{slots}. In every slot, the graph grows via two operations (i) vertex generation and (ii) edge activation. The process completes at the last slot where a (possibly empty) subset of the edges of the graph will be removed. Removed edges are called \emph{excess edges}. The main problem investigated in this paper is: Given a target graph $G$, we are asked to design an algorithm that outputs such a process growing $G$, called a \emph{growth schedule}. Additionally, the algorithm should try to minimize the total number of slots $k$ and of excess edges $\ell$ used by the process. We provide both positive and negative results for different values of $k$ and $\ell$, with our main focus being either schedules with sub-linear number of slots or with zero excess edges.