Irrelevant vertices for the planar Disjoint Paths Problem

Abstract

The DISJOINT PATHS PROBLEM asks, given a graph C and a set of pairs of terminals (s1, t1),... ,(sk, tk), whether there is a collection of k pairwise vertex-disjoint paths linking si and ti, for i = 1,.. . ,k. In their f(k) . n 3 algorithm for this problem, Robertson and Seymour introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(kΨ there is an "irrelevant" vertex whose removal creates an equivalent instance of the problem. This fact is based on the celebrated Unique Linkage Theorem, whose -very technical -proof gives a function g(k) that is responsible for an immense parameter dependence in the running time of the algorithm. In this paper we give a new and self-contained proof of this result that strongly exploits the combinatorial properties of planar graphs and achieves g(k) = O(k 3 / 2 . 2 k ). Our bound is radically better than the bounds known for general graphs.

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

“…We use the Flat Wall Theorem of Robertson and Seymour [44], in particular the recent optimized versions by Kawarabayashi et al [32] and by Chuzhoy [12]. In a nutshell, this theorem says that every K h -minor-free graph G has a set of vertices A ⊆ V (G) -called apices-with |A| = O h (1) such that G \ A contains a flat wall of height Ω h (tw(G)). Here, the definition of "flat wall" is quite involved and is detailed in Section 4; it essentially means a subgraph that has a bidimensional gridlike structure, separated from the rest of the graph by its perimeter, and that is "close" to being planar, in the sense that it can be embedded in the plane in a way that its potentially non-planar pieces, called flaps, have a well-defined structure along larger pieces called bricks.…”

“…The graph B e i,j is depicted in Figure 10. Informally, the graph B e i,j , for every e ∈ E(G), will play in F the role of the vertex (i, j) in G. For each e ∈ E(G) and each j ∈ [1, k], we define the graph C e j obtained from the disjoint union of every B e i,j , i ∈ [1, k], such that two graphs B e i 1 ,j and B e i 2 ,j , i 1 = i 2 , are complete to each other, that is, for every i 1…”

Hitting Topological Minor Models in Planar Graphs is Fixed Parameter Tractable

“…(3) ≤ is the subgraph or the induced subgraph relation: because of the result of Cai in [6], p -P F ,≤ -deletion is FPT, for every F. In particular, the result in [6] implies an O(h k n h+1 )-time algorithm for both these problems. However, if instead we parameterize P F ,≤ -deletion by h, 1 Let (x1, . .…”

A lower bound on the tree-width of graphs with irrelevant vertices

“…For planar graphs, an upper bound of f (k) ≤ 72 √ 2k 3 2 · 2 k was given in [1]. An elementary proof for a bound of f (k) ≤ (72k · 2 k − 72 · 2 k + 18) √ 2k + 1 was provided later [5] as well as a slightly improved bound of f (k) ≤ 26k · 2 3 2 · 2 k requiring a slightly more involved proof [2]. Our lower bound shows that this is asymptotically optimal.…”