  A Poisson outdegree-one graph is an oriented graph based on a Poisson point
process such that each vertex has only one outgoing edge. The paper focuses on
the absence of percolation for such graphs. Our main result is based on two
assumptions. The Shield assumption ensures that the graph is locally determined
with possible random horizons. The Loop assumption ensures that any forward
branch of the graph merges on a loop provided that the Poisson point process is
augmented with a finite collection of well-chosen points. Several models
satisfy these general assumptions and inherit in consequence the absence of
percolation. In particular, we solve a conjecture by Daley et al. on the
absence of percolation for the line-segment model. In this planar model, a
segment is growing from any point of the Poisson process and stops its growth
whenever it hits another segment. The random directions are picked
independently and uniformly on the unit sphere. Another model of geometric
navigation is presented and also fulfills the Shield and Loop assumptions.
