The self-replication introduced by John von Neumann is a process that produces a copy of itself. As a novel approach, this paper studies the self-replication process through the process of reproduction. In this paper, we propose a comprehensive graph reproduction system (GRS) and identify a specific reproduction system that turns out to be a graph self-replication system (GSS), with which a copy of any given graph can be produced through an algorithmic process. Unlike the GRS studied by Richard Southwell, our model considers the evolution of edges along with the evolution of vertices. We analyze some of the existing reproduction models through our system and identify the models that are self-replicable. © 2019, Complex Systems Publications, Inc. All rights reserved.