Broere, IzakHeidema, Johannes2020-09-282020-09-282020I. Broere and J. Heidema, Universality in graph properties allowing constrained growth, AKCE International Journal of Graphs and Combinatorics (2019), https://DOI.org/ 10.1016/j.akcej.2019.02.002. NYP.0972-8600 (print)2543-3474 (online)10.1016/j.akcej.2019.02.002http://hdl.handle.net/2263/76251A graph property is a class of graphs which is closed under isomorphisms. Some properties are also closed under one or more specified constructions that extend any graph into a supergraph containing the original graph as an induced subgraph.We introduce and study in particular the concept that a property P “allows finite spiking” and show that there is a universal graph in every induced-hereditary property of finite character which allows finite spiking. We also introduce the concept that P “allows isolated vertex addition” and constructively show that there is a unique graph with the so-called P-extension property in every induced-hereditary property P of finite character which allows finite spiking and allows isolated vertex addition; such a graph is then universal in P too. Infinitely many examples which satisfy the conditions of both these results are obtained by taking the property of Kn-free graphs for an arbitrary integer n ≥ 2.en© 2019 Kalasalingam University. This is an open access article under the CC BY-NC-ND license.Countable graphProperty of graphsUniversal graphFinite characterSpikingArticle