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Given a planar graph G on n vertices and an integer parameter r<n, anr-division of G with few holes is a decomposition of G into O(n/r) regions ofsize at most r such that each region contains at most a constant number offaces that are not faces of G (also called holes), and such that, for eachregion, the total number of vertices on these faces is O(sqrt r).  We provide a linear-time algorithm for computing r-divisions with few holes.In fact, our algorithm computes a structure, called decomposition tree, whichrepresents a recursive decomposition of G that includes r-divisions foressentially all values of r. In particular, given an exponentially increasingsequence r = (r_1,r_2,...), our algorithm can produce a recursive r-divisionwith few holes in linear time.  r-divisions with few holes have been used in efficient algorithms to computeshortest paths, minimum cuts, and maximum flows. Our linear-time algorithmimproves upon the decomposition algorithm used in the state-of-the-artalgorithm for minimum st-cut (Italiano, Nussbaum, Sankowski, and Wulff-Nilsen,STOC 2011), removing one of the bottlenecks in the overall running time oftheir algorithm (analogously for minimum cut in planar and bounded-genusgraphs).

Structured Recursive Separator Decompositions for Planar Graphs in Linear Time