Exploration on Highly Dynamic Graphs

TL;DR

This paper investigates the exploration problem in highly dynamic graphs, establishing new impossibility bounds and providing an algorithm that operates under specific visibility and memory constraints.

Contribution

It strengthens existing impossibility results for 1-Interval Connectivity and determines the minimal number of agents needed for exploration in Connectivity Time graphs.

Findings

Exploration is impossible with fewer than (n-1)(n-2)/2 agents in Connectivity Time models.

1-hop visibility is necessary for exploration with (n-1)(n-2)/2+1 agents.

An exploration algorithm is proposed using (n-1)(n-2)/2+1 agents with limited memory.

Abstract

We study the exploration problem by mobile agents in two prominent models of dynamic graphs: $1$-Interval Connectivity and Connectivity Time. The $1$-Interval Connectivity model was introduced by Kuhn et al.~[STOC 2010], and the Connectivity Time model was proposed by Michail et al.~[JPDC 2014]. Recently, Saxena et al.~[TCS 2025] investigated the exploration problem under both models. In this work, we first strengthen the existing impossibility results for the $1$-Interval Connectivity model. We then show that, in Connectivity Time dynamic graphs, exploration is impossible with $\frac{(n-1)(n-2)}{2}$ mobile agents, even when the agents have full knowledge of all system parameters, global communication, full visibility, and infinite memory. This significantly improves the previously known bound of $n$. Moreover, we prove that to solve exploration with $\frac{(n-1)(n-2)}{2}+1$ agents, $1$-hop visibility is necessary. Finally, we present an exploration algorithm that uses $\frac{(n-1)(n-2)}{2}+1$ agents, assuming global communication, $1$-hop visibility, and $O(\log n)$ memory per agent.