Exploration on Highly Dynamic Graphs

TL;DR

This work advances the understanding of graph exploration by mobile agents in highly dynamic networks. It delivers strong impossibility results for 1-Interval Connectivity and establishes a tight lower bound for Connectivity Time, showing that \\frac{(n-2)(n-1)}{2}\\) agents are insufficient even with unlimited resources, while \\frac{(n-2)(n-1)}{2}+1\\) agents suffice when agents have 1-hop visibility and global communication. The authors introduce a map-construction procedure (MAP) and a perpetual exploration algorithm (EXP_ALGO) that, together, achieve perpetual exploration with the stated agent count and memory bound \\mathcal{O}( )\\). The results illuminate fundamental trade-offs between visibility, communication, and memory in dynamic networks, with implications for robust deployment of autonomous agents in time-varying topologies.

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.

Figures (13)

• Figure 1: The construction of $G_1$ for $n=9$ and $p=6$.
• Figure 2: The construction of $G_2$ for $n=9$ and $p=6$.
• Figure 3: Initial configuration $\mathcal{C}_0$ of $\frac{(n-2)(n-1)}{2}$ agents.
• Figure 4: (A) Graph $\mathcal{G}_{iT-2}$, (B) Graph $\mathcal{G}_{iT-1}$.
• Figure 5: (A) An agent moves from node $w_{n-2}$ to node $w_{n-1}$ at round $r-1$ in $\mathcal{G}_{r-1}$, (B) $\mathcal{G}_{r}$ with respect to $\mathcal{G}_{r-1}$.

Theorems & Definitions (42)

• definition 1
• definition 2
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 5.1
• proof
• remark 1
• Lemma 5.1