## 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 \(f(k)\cdot n^{2}\).

## Keywords

Completion problems Disjoint paths Planar graphs## Notes

### Acknowledgments

We wish to thank the anonymous reviewers of an earlier version of this paper for valuable comments and suggestions.

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