Planar Disjoint-Paths Completion

Abstract

We introduce Planar Disjoint Paths Completion, a completion counterpart of the Disjoint Paths problem, and study its parameterized complexity. The problem can be stated as follows: given a, not necessarily connected, plane graph G, k pairs of terminals, and a face F of G, find a minimum-size set of edges, if one exists, to be added inside F so that the embedding remains planar and the pairs become connected by k disjoint paths in the augmented network. Our results are twofold: first, we give an upper bound on the number of necessary additional edges when a solution exists. This bound is a function of k, independent of the size of G. Second, we show that the problem is fixed-parameter tractable, in particular, it can be solved in time f (k) · n 2 .

A survey of parameterized algorithms and the complexity of edge modification

“…Adler et al [3] introduced the problem of augmenting a planar graph G with a given set of k pairs of terminals. The task is to augment G with the minimum number of edges such that all edges are added within one face of G, the augmented graph is planar and all terminal-pairs are linked with vertex-disjoint paths.…”

Irrelevant vertices for the planar Disjoint Paths Problem

“…This naturally partitions the vertices of bnd(D)∩L into the up and down ones. The following proposition is implicit in the proof of Theorem 2 in [3] (see the derivation of the unique claim in the proof of the former theorem). See also [8] for related results.…”

A polynomial-time algorithm for Outerplanar Diameter Improvement

“…Such a completion problem is the Planar Disjoint Paths Completion problem that asks, given a plane graph and a collection of k pairs of terminals, whether it is possible to add edges such that the resulting graph remains plane and contains k vertex-disjoint paths between the pairs of terminals. While this problem is trivially NP-complete, it has been studied from the point of view of parameterized complexity [1]. In particular, when all edges should be added in the same face, it can be solved in f (k) • n 2 steps [1], i.e., it is fixed parameter tractable (FPT in short).…”

“…While this problem is trivially NP-complete, it has been studied from the point of view of parameterized complexity [1]. In particular, when all edges should be added in the same face, it can be solved in f (k) • n 2 steps [1], i.e., it is fixed parameter tractable (FPT in short).…”

“…The computational complexity of Planar Diameter Improvement is open, as it is not even known whether it is an NP-complete problem, even in the case where the embedding is part of the input. Interestingly, Planar Diameter Improvement is known to be FPT: it is easy to verify that, for every D, its Yes-instances are closed under taking minors 1 which, according to the meta-algorithmic consequence of the Graph Minors series of Robertson and Seymour [23,24], implies that Planar Diameter Improvement is FPT. Unfortunately, this implication only proves the existence of such an algorithm for each D, while it does not give any way to construct it.…”