  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 generalised 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.
