Abstract
We study the effects on graph width parameters of planarization, the construction of a planar diagram from a non-planar graph drawing by replacing each crossing with a new vertex. 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. To show our lower bounds on the width of planarizations, we prove that arrangements of curves with many crossing pairs of curves must generate planar graphs of high width.
Highlights
Planarization is a graph transformation, standard in graph drawing, in which a given graph G is drawn in the plane with simple crossings of pairs of edges, and each crossing of two edges in the drawing is replaced by a new dummy vertex, subdividing the two edges [1–4]
This should be distinguished from a different problem, called planarization, in which we try to find a large planar subgraph of a nonplanar graph [5–8]
One of their constructions involved the planarization of a nonplanar graph of bounded pathwidth, and they observed that the planarization maintained the low pathwidth of their graph
Summary
Planarization is a graph transformation, standard in graph drawing, in which a given graph G is drawn in the plane with simple crossings of pairs of edges, and each crossing of two edges in the drawing is replaced by a new dummy vertex, subdividing the two edges [1–4] This should be distinguished from a different problem, called planarization, in which we try to find a large planar subgraph of a nonplanar graph [5–8]. The size of the planarization (equivalently the crossing number of G) is of primary importance in graph drawing, it is natural to ask what other properties can be transferred from G to its planarizations One problem of this type arose in the work of Jansen and Wulms on the fixed-parameter tractability of graph optimization problems on graphs of bounded pathwidth [9]. In contrast, we are assuming that the original graph has low width and we derive properties of its planarization from that assumption
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