TY - GEN
T1 - Path Connected Dynamic Graphs with a Study of Efficient Dispersion
AU - Saxena, Ashish
AU - Mondal, Kaushik
N1 - Publisher Copyright:
© 2025 Copyright held by the owner/author(s).
PY - 2025/1/4
Y1 - 2025/1/4
N2 - In dynamic graphs, usually, several edges may get added or deleted in a round with respect to the previous round. There are different connectivity models based on the constraints on the addition/deletion of edges. One such model is known as 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 for dynamic graphs, namely 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. According to 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. We show that our model is weaker than T-Interval Connectivity but stronger than Connectivity Time model. We study the dispersion problem of mobile agents on our connectivity model. Dispersion is already studied in the T-Interval Connectivity model for T = 1. We show that the existing algorithm in 1-Interval Connected graphs for dispersion with termination does not work in our 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 and memory with those minimal model assumptions on T-Path Connected dynamic graphs. Also, we show that solving dispersion is impossible in the Connectivity Time model even in the presence of several other strong model assumptions. This further exhibits that the Connectivity Time model is indeed the weakest model among these three models. We believe other problems like exploration, gathering can be studied in our T-Path Connectivity model.
AB - In dynamic graphs, usually, several edges may get added or deleted in a round with respect to the previous round. There are different connectivity models based on the constraints on the addition/deletion of edges. One such model is known as 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 for dynamic graphs, namely 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. According to 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. We show that our model is weaker than T-Interval Connectivity but stronger than Connectivity Time model. We study the dispersion problem of mobile agents on our connectivity model. Dispersion is already studied in the T-Interval Connectivity model for T = 1. We show that the existing algorithm in 1-Interval Connected graphs for dispersion with termination does not work in our 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 and memory with those minimal model assumptions on T-Path Connected dynamic graphs. Also, we show that solving dispersion is impossible in the Connectivity Time model even in the presence of several other strong model assumptions. This further exhibits that the Connectivity Time model is indeed the weakest model among these three models. We believe other problems like exploration, gathering can be studied in our T-Path Connectivity model.
KW - Anonymous graphs
KW - Deterministic algorithm
KW - Dispersion
KW - Dynamic graphs
KW - Mobile agents
UR - http://www.scopus.com/inward/record.url?scp=85218347528&partnerID=8YFLogxK
U2 - 10.1145/3700838.3700864
DO - 10.1145/3700838.3700864
M3 - Conference contribution
AN - SCOPUS:85218347528
T3 - ICDCN 2025 - Proceedings of the 26th International Conference on Distributed Computing and Networking
SP - 171
EP - 180
BT - ICDCN 2025 - Proceedings of the 26th International Conference on Distributed Computing and Networking
PB - Association for Computing Machinery, Inc
T2 - 26th International Conference on Distributed Computing and Networking, ICDCN 2025
Y2 - 4 January 2025 through 7 January 2025
ER -