Dynamic parameterized problems on unit disk graphs

TL;DR

This paper introduces the first dynamic data structures for key parameterized problems on unit disk graphs, achieving efficient update and query times, which is a significant advancement in geometric graph algorithms.

Contribution

It presents novel data structures for dynamic parameterized problems on unit disk graphs, with specific efficiency guarantees for various problems.

Findings

Supports $2^{O(\sqrt{k})}$ update time for $k$-Path/Cycle.

Supports $O(\log n)$ update time for other problems.

Achieves efficient dynamic algorithms for problems previously unaddressed on unit disk graphs.

Abstract

In this paper, we study fundamental parameterized problems such as $k$-Path/Cycle, Vertex Cover, Triangle Hitting Set, Feedback Vertex Set, and Cycle Packing for dynamic unit disk graphs. Given a vertex set $V$ changing dynamically under vertex insertions and deletions, our goal is to maintain data structures so that the aforementioned parameterized problems on the unit disk graph induced by $V$ can be solved efficiently. Although dynamic parameterized problems on general graphs have been studied extensively, no previous work focuses on unit disk graphs. In this paper, we present the first data structures for fundamental parameterized problems on dynamic unit disk graphs. More specifically, our data structure supports $2^{O(\sqrt{k})}$ update time and $O(k)$ query time for $k$-Path/Cycle. For the other problems, our data structures support $O(\log n)$ update time and $2^{O(\sqrt{k})}$ query time, where $k$ denotes the output size.