Transitive closure in a polluted environment

We introduce and study a new percolation model, inspired by recent works on jigsaw percolation, graph bootstrap in polluted environments. Start with an oriented G0 of initially occupied edges n vertices, iteratively occupy additional (oriented) transitivity, the constraint that only open certain random set can ever be occupied. All other are closed, creating obstacles for spread edges. When is unoriented linear graph, leftward rightward independently possibly different probabilities, we identify three regimes which eventually either all edges, majority one direction, or very small proportion In more general setting where connected bounded degree, show transition between sparse full occupation occurs when probability (logn)−1/2+o(1). conclude several conjectures problems.