Variants of Plane Diameter Completion

TL;DR

This paper studies two variants of the Plane Diameter Completion problem, proving their NP-completeness and providing parameterized algorithms with double-exponential complexity in certain parameters.

Contribution

It introduces two new variants of the problem, proves their NP-completeness under specific conditions, and offers parameterized algorithms for both.

Findings

Both variants are NP-complete even under restricted graph conditions.

Parameterized algorithms run in polynomial plus double-exponential time.

The problems remain hard even with tight constraints on added edges.

Abstract

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.