Hitting Minors on Bounded Treewidth Graphs. I. General Upper Bounds

Abstract

For a finite collection of graphs F , the F-M-Deletion problem consists in, given a graph G and an integer k, deciding whether there exists S ⊆ V (G) with |S| ≤ k such that G \ S does not contain any of the graphs in F as a minor. We are interested in the parameterized complexity of F-M-Deletion when the parameter is the treewidth of G, denoted by tw. Our objective is to determine, for a fixed F , the smallest function f F such that F-M-Deletion can be solved in time f F (tw) • n O(1) on n-vertex graphs. We prove that f F (tw) = 2 2 O(tw•log tw) for every collection F , that f F (tw) = 2 O(tw•log tw) if F contains a planar graph, and that f F (tw) = 2 O(tw) if in addition the input graph G is planar or embedded in a surface. We also consider the version of the problem where the graphs in F are forbidden as topological minors, called F-TM-Deletion. We prove similar results for this problem, except that in the last two algorithms, instead of requiring F to contain a planar graph, we need it to contain a subcubic planar graph. This is the first of a series of articles on this topic.

Hitting minors on bounded treewidth graphs. III. Lower bounds

“…Note that, by definition, it holds that C ⊆ Q, but we consider both sets because we will prove a stronger result for the set C (Theorem 3, which applies to every subset of C) than for the set Q (Theorems 4 and 5, which apply to families containing a single graph H). 1) by the algorithms presented in [5][6][7]. In particular, note that these results altogether settle completely the asymptotic complexity of {H}-M-Deletion when H is a connected graph; see Figure 1 for an illustration.…”

“…Results in other articles of the series and discussion. In the first article of this series [7], we show, among other results, that for every connected F containing at least one planar graph (resp. subcubic planar graph), F-M-Deletion (resp.…”

“…Concerning the topological minor version, in order to establish a dichotomy for {H}-TM-Deletion when H is planar and connected, it remains to obtain algorithms in time 2 O(tw•log tw) • n O (1) for the graphs H with maximum degree at least four, like the gem or the dart (see Figure 1), as for those graphs the algorithm in time 2 O(tw•log tw) • n O (1) given in [7] cannot be applied.…”

Hitting minors on bounded treewidth graphs. II. Single-exponential algorithms

“…Let G be a connected graph with |V (G)| ≥ 5. It is straightforward to see that if G is a cycle or if C 4 tm G, then banner tm G. Conversely, assume that G excludes the Concerning the topological minor version, in order to establish a dichotomy for {H}-TM-Deletion when H is planar and connected, it remains to obtain algorithms in time O * (2 O(tw•log tw) ) for the graphs H with maximum degree at least four, like the gem or the dart (see Figure 1), as for those graphs the algorithm in time O * (2 O(tw•log tw) ) given in [6] cannot be applied.…”

“…This article triggered several other techniques to obtain single-exponential deterministic algorithms for so-called connectivity problems on graphs of bounded treewidth, mostly based on algebraic tools [9,18]. We refer the reader to [6] for a more detailed discussion about related work. In particular, in this article we make use of one of the techniques presented by Bodlaender et al [9], called rank-based approach.…”