  We introduce and study a new percolation model, inspired by recent works on
jigsaw percolation, graph bootstrap percolation, and percolation in polluted
environments. Start with an oriented graph $G_0$ of initially occupied edges on
$n$ vertices, and iteratively occupy additional (oriented) edges by
transitivity, with the constraint that only open edges in a certain random set
can ever be occupied. All other edges are closed, creating a set of obstacles
for the spread of occupied edges. When $G_0$ is an unoriented linear graph, and
leftward and rightward edges are open independently with possibly different
probabilities, we identify three regimes in which the set of eventually
occupied edges is either all open edges, the majority of open edges in one
direction, or only a very small proportion of all open edges. In the more
general setting where $G_0$ is a connected unoriented graph of bounded degree,
we show that the transition between sparse and full occupation of open edges
occurs when the probability of open edges is $(\log n)^{-1/2+o(1)}$. We
conclude with several conjectures and open problems.
