article published yes Sofiya Burova author Lyuben Lichev author 9aa8388e-d003-11ee-8458-c4c1d7447977 MaKw department The online semi-random graph process is a one-player game which starts with the empty graph on n vertices. At every round, a player (called Builder) is presented with a vertex v chosen uniformly at random and independently from previous rounds, and constructs an edge of their choice that is incident to v. Inspired by recent advances on the semi-random graph process, we define a family of generalized online semi-random models. We analyse a particular instance that shares similar features with the original semi-random graph process and determine the hitting times of the classical graph properties minimum degree k,k-connectivity, containment of a perfect matching, a Hamiltonian cycle and an H-factor for a fixed graph H possessing an additional tree-like property. Along the way, we derive a few consequences of the famous Aldous-Broder algorithm that may be of independent interest. Elsevier2025 eng 0195-6698 2204.07376 00142065940000110.1016/j.ejc.2025.104120 126 S. Burova, L. Lichev, European Journal of Combinatorics 126 (2025). Burova, Sofiya, and Lyuben Lichev. “The Semi-Random Tree Process.” <i>European Journal of Combinatorics</i>. Elsevier, 2025. <a href="https://doi.org/10.1016/j.ejc.2025.104120">https://doi.org/10.1016/j.ejc.2025.104120</a>. Burova S, Lichev L. 2025. The semi-random tree process. European Journal of Combinatorics. 126, 104120. Burova, Sofiya, and Lyuben Lichev. “The Semi-Random Tree Process.” <i>European Journal of Combinatorics</i>, vol. 126, 104120, Elsevier, 2025, doi:<a href="https://doi.org/10.1016/j.ejc.2025.104120">10.1016/j.ejc.2025.104120</a>. Burova S, Lichev L. The semi-random tree process. <i>European Journal of Combinatorics</i>. 2025;126. doi:<a href="https://doi.org/10.1016/j.ejc.2025.104120">10.1016/j.ejc.2025.104120</a> Burova, S., &#38; Lichev, L. (2025). The semi-random tree process. <i>European Journal of Combinatorics</i>. Elsevier. <a href="https://doi.org/10.1016/j.ejc.2025.104120">https://doi.org/10.1016/j.ejc.2025.104120</a> S. Burova and L. Lichev, “The semi-random tree process,” <i>European Journal of Combinatorics</i>, vol. 126. Elsevier, 2025. 190182025-02-10T09:00:53Z2025-09-30T10:28:42Z