  We define a special case of tree decompositions for planar graphs that
respect a given embedding of the graph. We study the analogous width of the
resulting decomposition we call the embedded-width of a plane graph. We show
both upper bounds and lower bounds for the embedded-width of a graph in terms
of its treewidth and describe a fixed parameter tractable algorithm to
calculate the embedded-width of a plane graph. To do so, we give novel bounds
on the size of matchings in planar graphs.
