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We study the random planar graph process introduced by Gerke, Schlatter,Steger, and Taraz [The random planar graph process, Random StructuresAlgorithms 32 (2008), no. 2, 236--261; MR2387559]: Begin with an empty graph on$n$ vertices, consider the edges of the complete graph $K_n$ one by one in arandom ordering, and at each step add an edge to a current graph only if thegraph remains planar. They studied the number of edges added up to step $t$ for'large' $t=\omega (n)$. In this paper we extend their results by determiningthe asymptotic number of edges added up to step $t$ in the early evolution ofthe process when $t=O(n)$. We also show that this result holds for a much moregeneral class of graphs, including outerplanar graphs, planar graphs, andgraphs on surfaces.

The early evolution of the random graph process in planar graphs and related classes