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We define a special case of tree decompositions for planar graphs thatrespect a given embedding of the graph. We study the analogous width of theresulting decomposition we call the embedded-width of a plane graph. We showboth upper bounds and lower bounds for the embedded-width of a graph in termsof its treewidth and describe a fixed parameter tractable algorithm tocalculate the embedded-width of a plane graph. To do so, we give novel boundson the size of matchings in planar graphs.

Embedded-width: A variant of treewidth for plane graphs