Dynamic domination and independence in sparse graphs

2026-07-24

AI Summary

This work addresses the dynamic existence queries for distance-$r$ dominating sets and independent sets in sparse graphs. Leveraging structural properties such as bounded expansion and bounded degeneracy, the paper introduces a novel approach that integrates randomized techniques with dynamic data structures. For graph classes with bounded expansion or constant degeneracy, the proposed method achieves the first sublogarithmic amortized update time: $\log^c n \cdot \log(1/\varepsilon)$ in general, and an improved bound of $2^{k^{O(d)}} \cdot \log^3 n \cdot \log(1/\varepsilon)$ for $d$-degenerate graphs. Additionally, it presents a $d^{O(1)} \cdot \log n$ update-time algorithm for maintaining an $O(d^2)$-approximate dominating set, significantly advancing the state of the art in dynamic graph algorithms for sparse networks.

Abstract

Let $\mathscr{C}$ be a class of graphs of bounded expansion and $r,k\in \mathbb{N}$ be fixed. We give a dynamic data structure that for a given dynamic graph $G$, updated by edge insertions and deletions subject to the promise that $G\in \mathscr{C}$ at all times, maintains the answer to the following two queries: (a) Does $G$ contain a distance-$r$ dominating set of size $k$? (b) Does $G$ contain a distance-$r$ independent set of size $k$? The data structure is randomized with error probability bounded by $\varepsilon$, for a parameter $\varepsilon>0$ fixed upon the initialization. The amortized update time is $\log^c n\cdot \log \frac{1}{\varepsilon}$, where $n$ is the vertex count of $G$ and $c$ is a constant that depends only on $r$, $k$, and $\mathscr{C}$. In the case of the first query, the data structure can also output a distance-$r$ dominating set of size $k$, if existent. We also prove that when $r=1$, our data structure for the dominating set query can be implemented even if we only assume that the maintained graph $G$ has degeneracy bounded by a constant $d$, yielding a simpler data structure with an improved amortized update time of $2^{k^{{\cal O}(d)}}\cdot \log^3 n\cdot \log \frac{1}{\varepsilon}$. Finally, we prove that in graphs of degeneracy at most $d$, one can maintain an ${\cal O}(d^2)$-approximation of the minimum size of a (distance-$1$) dominating set with amortized expected update time $d^{{\cal O}(1)}\cdot \log n$.

Problem

• dynamic graphs
• distance-r domination
• distance-r independence
• sparse graphs
• bounded expansion

Innovation

• dynamic data structure
• bounded expansion
• distance-r domination
• degeneracy
• amortized update time