A Near-Optimal Planarization Algorithm

Abstract

The problem of testing whether a graph is planar has been studied for over half a century, and is known to be solvable in O(n) time using a myriad of different approaches and techniques. Robertson and Seymour established the existence of a cubic algorithm for the more general problem of deciding whether an n-vertex graph can be made planar by at most k vertex deletions, for every fixed k. Of the known algorithms for k-Vertex Planarization, the algorithm of Marx and Schlotter (WG 2007, Algorithmica 2012) running in time 22 achieves the best running time dependence on k. The algorithm ofthat is not stated explicitly, achieves the best dependence on n.In this paper we present an algorithm for k-Vertex Planarization with running time 2 O(k log k) · n, significantly improving the running time dependence on k without compromising the linear dependence on n. Our main technical contribution is a novel scheme to reduce the treewidth of the input graph to O(k) in time 2 O(k log k) · n. It combines new insights into the structure of graphs that become planar after contracting a matching, with a Baker-type subroutine that reduces the number of disjoint paths through planar parts of the graph that are not affected by the sought solution. To solve the reduced instances we formulate a dynamic programming algorithm for Weighted Vertex Planarization on graphs of treewidth w with running time 2 O(w log w) · n, thereby improving over previous double-exponential algorithms.While Kawarabayashi's planarization algorithm relies heavily on deep results from the graph minors project, our techniques are elementary and practically self-contained. We expect them to be applicable to related edge-deletion and contraction variants of planarization problems.

A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary

“…Informally, here "irrelevant" means a non-boundary vertex of R that can be avoided by any minor model of a graph on at most h vertices and edge that traverses the boundary of R, no matter the graph that may be glued to it and no matter how this model traverses the boundary of R; see Section 5 for the precise definition. The irrelevant vertex technique originated in the seminal work of Robertson and Seymour [44,45] and has become a very useful tool used in various kinds of linkage and cut problems [1,28,35,36,42]. Nevertheless, given the nature of our setting, it is critical that the size of the flat wall where the irrelevant vertex appears does not depend of the boundary size.…”

When Recursion is Better than Iteration: A Linear-Time Algorithm for Acyclicity with Few Error Vertices

“…Then, a linear-time FPT algorithm for the compression version is executed to solve the general problem on this instance. Some of the results that fall under this paradigm are Bodlaender's linear FPT algorithm for Treewidth [3], the FPT-approximation algorithms for Treewidth [4,44], as well as algorithms for Vertex Planarization [25,33]. Let us call this the method of recursive compression.…”

“…It was only relatively recently that the first linear time algorithms were obtained for testing whether a graph is k vertices away from being planar [33,25] or bipartite [31,43]. Some of the other important results in this line of research include the linear time algorithms for Subgraph Isomorphism [12], Subset Feedback Vertex Set [37], Planar F-Deletion [2,3,16,19,18], Crossing Number [22,23,28], Interval Vertex Deletion [6], as well as a single-exponential and linear time parameterized constant factor approximation algorithm for Treewidth [4].…”

On the number of labeled graphs of bounded treewidth

“…When designing a parameterized algorithm, usually a crucial step is to solve the problem at hand restricted to graphs decomposable along small separators by performing dynamic programming (see [14] for a recent example). For instance, precise bounds on T n,k are useful when dealing with the Treewidth-k Vertex Deletion problem, which has recently attracted significant attention in the area [9,10,15].…”

“…From Equation (14) and using that an n-vertex labeled proper linear k-tree has kn − k(k+1) 2 edges, basic calculations yield that the dominant term of the number of n-vertex labeled graphs of proper-pathwidth at most k is at most k·2 k ·n c n for some absolute constant c ≥ 1.88. Finally, it would be interesting to count the graphs of bounded Xwidth, for other X different than "tree", "path", or "proper-path".…”