  We consider decomposing a 3-connected planar graph $G$ using laminar
separators of size three. We show how to find a maximal set of laminar
3-separators in such a graph in linear time. We also discuss how to find
maximal laminar set of 3-separators from special families. For example we
discuss non-trivial cuts, ie. cuts which split $G$ into two components of size
at least two. For any vertex $v$, we also show how to find a maximal set of
3-separators disjoint from $v$ which are laminar and satisfy: every vertex in a
separator $X$ has two neighbours not in the unique component of $G-X$
containing $v$. In all cases, we show how to construct a corresponding tree
decomposition of adhesion three. Our new algorithms form an important component
of recent methods for finding disjoint paths in nonplanar graphs.
