  We consider undirected graphs that arise as deterministic functions of
stationary point processes such that each point has degree bounded by two. For
a large class of point processes and edge-drawing rules, we show that the
arising graph has no infinite connected component, almost surely. In
particular, this extends our previous result for SINR graphs based on
stabilizing Cox point processes and verifies the conjecture of Balister and
Bollob\'as that the bidirectional $k$-nearest neighbor graph of a
two-dimensional homogeneous Poisson point process does not percolate for $k=2$.
