  We study the effects of planarization (the construction of a planar diagram
$D$ from a non-planar graph $G$ by replacing each crossing by a new vertex) on
graph width parameters. We show that for treewidth, pathwidth, branchwidth,
clique-width, and tree-depth there exists a family of $n$-vertex graphs with
bounded parameter value, all of whose planarizations have parameter value
$\Omega(n)$. However, for bandwidth, cutwidth, and carving width, every graph
with bounded parameter value has a planarization of linear size whose parameter
value remains bounded. The same is true for the treewidth, pathwidth, and
branchwidth of graphs of bounded degree.
