The complexity of growing a graph

We study a new algorithmic process of graph growth which starts from a single initial vertex and operates in discrete time-steps, called 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 are removed. Removed edges are called excess edges. The main problem investigated in this paper is: Given a target graph G, design an algorithm that outputs a process that grows G, called a growth schedule. Additionally, we aim to minimize the total number of slots k and of excess edges ℓ used by the process. We provide both positive and negative results, with our main focus being either schedules with sub-linear number of slots or with no excess edges.