Path connected dynamic graphs with a study of dispersion and exploration
In dynamic graphs, several edges may get added or deleted in a round. There are different connectivity models based on the constraints on the addition/deletion of edges. One such model is the T-Interval Connectivity model, where edges can be added/deleted, keeping the graph nodes connected in each s...
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| Veröffentlicht in: | Theoretical computer science Jg. 1050; S. 115390 |
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| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Elsevier B.V
27.09.2025
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| Schlagworte: | |
| ISSN: | 0304-3975 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | In dynamic graphs, several edges may get added or deleted in a round. There are different connectivity models based on the constraints on the addition/deletion of edges. One such model is the T-Interval Connectivity model, where edges can be added/deleted, keeping the graph nodes connected in each synchronous round. The parameter T depends on the stability of the underlying connected structure across rounds. There is another connectivity model, namely the Connectivity Time model, where the union of all the edges present in any T consecutive rounds must form a connected graph. This is much weaker than the T-Interval Connectivity as the graph may even be disconnected at each round. We, in this work, come up with a new connectivity model, namely T-Path Connectivity. In our model, the nodes may not remain connected in each round, but for any pair of nodes u,v, there must exist path(s) at least once in any consecutive T rounds. Our model is weaker than T-Interval Connectivity but stronger than the Connectivity Time model.
We study the dispersion problem in our connectivity model. Dispersion is already studied in the 1-Interval Connectivity model. We show that the existing algorithm in 1-Interval Connected graphs for dispersion with termination does not work in our T-Path Connectivity model for obvious reasons. We answer what are the necessary assumptions to solve dispersion in our connectivity model. Then, we provide an algorithm that runs in optimal time with those minimal model assumptions on T-Path Connected graphs. Also, we show that solving dispersion is impossible in the Connectivity Time model, even in the presence of several other strong model assumptions. Further, we initiate the study of the exploration problem on these three connectivity models. We provide several impossibility results with different assumptions. In most cases, we establish necessary and sufficient conditions to solve the exploration problem using an optimal number of agents in an asymptomatically optimal time. It is also evident from the studies of dispersion as well as exploration on all the three connectivity models that, Connectivity Time model is indeed the weakest model among these three models.
•Introduce a connectivity model for dynamic graphs, namely, T-Path Connectivity.•Necessary conditions to solve Implicit/Explicit dispersion on different connectivity models for dynamic graphs along with some impossibility results.•Time optimal algorithm to solve dispersion.•Provide necessary and sufficient conditions to solve the exploration problem for different connectivity models along with some impossibility results.•Time optimal algorithm to solve exploration except for Connectivity Time model. |
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| ISSN: | 0304-3975 |
| DOI: | 10.1016/j.tcs.2025.115390 |