Recursive Separator Decompositions for Planar Graphs

Years and Authors of Summarized Original Work

• 1987; Frederickson

• 1995; Goodrich

• 2013; Klein, Mozes, Sommer

Problem Definition

Graph decompositions are the basis for many divide-and-conquer algorithms. Two main properties make a decomposition useful. The first is balance, namely, that the parts of the decomposition have roughly the same size. Balanced decompositions lead to logarithmic depth recursion. The second is small overlap between the parts of the decomposition. The overlap affects the time it takes to combine solutions of different parts into a solution for the union of the parts.

A decomposition of a graph G is a collection of subgraphs of G, called regions, whose union is G. A decomposition tree of G is a tree $\mathcal{T}$ whose nodes correspond to subgraphs of G. The root of $\mathcal{T}$ consists of the entire graph G. For a node v of $\mathcal{T}$ that corresponds to a subgraph R, the children $v_{1},v_{2},\ldots,v_{k}$ of v correspond to subgraphs of Rwhose...