Planar Disjoint-Paths Completion

TL;DR

This paper introduces the Planar Disjoint Paths Completion problem, studying its parameterized complexity, providing bounds on additional edges needed, and proving it is fixed-parameter tractable with respect to the number of terminal pairs.

Contribution

It defines a new graph completion problem related to disjoint paths in planar graphs and establishes fixed-parameter tractability results.

Findings

Bound on the number of additional edges needed depends only on k.

The problem is fixed-parameter tractable with respect to k.

The algorithm runs in time f(k)·n^2.

Abstract

introduce {\sc 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 $f(k)\cdot n^{2}.$