Gathering in 1-Interval Connected Graphs

TL;DR

This paper investigates the problem of multi-agent gathering in dynamic, always-connected graphs, identifying impossibility conditions and proposing a polynomial-time algorithm for weak gathering in unicyclic graphs.

Contribution

It characterizes the solvability of gathering in 1-interval connected graphs and introduces a deterministic algorithm for weak gathering in unicyclic graphs.

Findings

Gathering is impossible in graphs with cycles.

Weak gathering is solvable in unicyclic graphs.

A polynomial-time algorithm for weak gathering in unicyclic graphs is provided.

Abstract

We examine the problem of gathering $k \geq 2$ agents (or multi-agent rendezvous) in dynamic graphs which may change in every synchronous round but remain always connected ($1$-interval connectivity) [KLO10]. The agents are identical and without explicit communication capabilities, and are initially positioned at different nodes of the graph. The problem is for the agents to gather at the same node, not fixed in advance. We first show that the problem becomes impossible to solve if the graph has a cycle. In light of this, we study a relaxed version of this problem, called weak gathering. We show that only in unicyclic graphs weak gathering is solvable, and we provide a deterministic algorithm for this problem that runs in polynomial number of rounds.