Path Connected Dynamic Graphs with a Study of Dispersion and Exploration

TL;DR

This work introduces $T$-Path Connectivity, a dynamic-graph connectivity model strictly between $T$-Interval Connectivity and Connectivity Time. It shows dispersion is solvable in $T$-Path Connected graphs with optimal $O(kT)$ time under minimal assumptions, while dispersion remains impossible in Connectivity Time. For exploration, the paper establishes tight lower bounds and provides optimal algorithms across the three models, including $igr| ext{Θ}(n)igr|$ rounds for 1-Interval Connected exploration and $igr| ext{Θ}(nT)igr|$ rounds for $T$-Path Connected exploration with $n-1$ agents. Overall, Connectivity Time is the weakest model among the three, while $T$-Path Connectivity offers a practical middle ground enabling robust dispersion and exploration under adversarial dynamics.

Abstract

In dynamic graphs, edges may be added or deleted in each synchronous round. Various connectivity models exist based on constraints on these changes. One well-known model is the $T$-Interval Connectivity model, where the graph remains connected in every round, and the parameter $T$ reflects the duration of structural stability. Another model is Connectivity Time, where the union of edges across any $T$ consecutive rounds forms a connected graph. This is a weaker model, as the graph may be disconnected in individual rounds. In this work, we introduce a new connectivity model called $T$-Path Connectivity. Unlike $T$-Interval Connectivity, the graph may not be connected in each round, but for every pair of nodes $u,v$, there must exist a path connecting them in at least one round within any $T$ consecutive rounds. This model is strictly weaker than $T$-Interval Connectivity but stronger than the Connectivity Time model. We study the dispersion problem in the $T$-Path Connectivity model. While dispersion has been explored in the 1-Interval Connectivity model, we show that the existing algorithm with termination does not work in our model. We then identify the minimal necessary assumptions required to solve dispersion in the $T$-Path Connectivity model and provide an algorithm that solves it optimally under those conditions. Moreover, we prove that dispersion is unsolvable in the Connectivity Time model, even under several strong assumptions. We further initiate the study of the exploration problem under all three connectivity models. We present multiple impossibility results and, in most cases, establish tight bounds on the number of agents and time required. Our results demonstrate that, in both dispersion and exploration, the Connectivity Time model is strictly the weakest among the three.

Figures (21)

• Figure 1: (a) Graph $\mathcal{G}_r$, where $r$(mod 3)=0, (b) Graph $\mathcal{G}_{r}$, where $r$(mod 3)=1, (c) Graph $\mathcal{G}_{r}$, where $r$(mod 3)=2. This figure is an example of $T$-Path Connectivity for $T=3$.
• Figure 2: (a) Graph $\mathcal{G}_r$, where $r$(mod 3)=0, (b) Graph $\mathcal{G}_{r}$, where $r$(mod 3)=1, (c) Graph $\mathcal{G}_{r}$, where $r$(mod 3)=2. This figure is an example of Connectivity Time for $T=3$.
• Figure 3: (a) $\mathcal{G}$ for $n=7$ with agents' position, (b) $\mathcal{G'}$ for $n=7$ with agents' position
• Figure 4: (a) $\mathcal{G}_{t+1}$ for $n=7$ with agents' position, (b) $\mathcal{G}_r$, $r>t+1$, for $n=7$ with agents' position
• Figure 5: (a) $\mathcal{G}_r$ at round $r$, where $p(T-1)+1 \leq r < (p+1)(T-1)$, (b) $\mathcal{G}_r$ at round $r$, where $r=(p+1)(T-1)$

Theorems & Definitions (53)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Theorem 5.1
• Theorem 5.2
• Theorem 5.3