Planar Disjoint-Paths Completion

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) · n2

A Polynomial-time Algorithm for Outerplanar Diameter Improvement

The Outerplanar Diameter Improvement problem asks, given a graph $G$ and an integer $D$, whether it is possible to add edges to $G$ in a way that the resulting graph is outerplanar and has diameter at most $D$. We provide a dynamic programming algorithm that solves this problem in polynomial time. Outerplanar Diameter Improvement demonstrates several structural analogues to the celebrated and challenging Planar Diameter Improvement problem, where the resulting graph should, instead, be planar. The complexity status of this latter problem is open.Comment: 24 page

Variants of Plane Diameter Completion

The {\sc Plane Diameter Completion} problem asks, given a plane graph $G$ and a positive integer $d$, if it is a spanning subgraph of a plane graph $H$ that has diameter at most $d$. We examine two variants of this problem where the input comes with another parameter $k$. In the first variant, called BPDC, $k$ upper bounds the total number of edges to be added and in the second, called BFPDC, $k$ upper bounds the number of additional edges per face. We prove that both problems are {\sf NP}-complete, the first even for 3-connected graphs of face-degree at most 4 and the second even when $k=1$ on 3-connected graphs of face-degree at most 5. In this paper we give parameterized algorithms for both problems that run in $O(n^{3})+2^{2^{O((kd)^2\log d)}}\cdot n$ steps.Comment: Accepted in IPEC 201

Lack of Sphere Packing of Graphs via Non-Linear Potential Theory

It is shown that there is no quasi-sphere packing of the lattice grid Z^{d+1} or a co-compact hyperbolic lattice of H^{d+1} or the 3-regular tree \times Z, in R^d, for all d. A similar result is proved for some other graphs too. Rather than using a direct geometrical approach, the main tools we are using are from non-linear potential theory.Comment: 10 page