On the Treewidth of Dynamic Graphs
Dynamic graph theory is a novel, growing area that deals with graphs that change over time and is of great utility in modelling modern wireless, mobile and dynamic environments. As a graph evolves, possibly arbitrarily, it is challenging to identify the graph properties that can be preserved over time and understand their respective computability.
In this paper we are concerned with the treewidth of dynamic graphs. We focus on metatheorems, which allow the generation of a series of results based on general properties of classes of structures. In graph theory two major metatheorems on treewidth provide complexity classifications by employing structural graph measures and finite model theory. Courcelle’s Theorem gives a general tractability result for problems expressible in monadic second order logic on graphs of bounded treewidth, and Frick & Grohe demonstrate a similar result for first order logic and graphs of bounded local treewidth.
We extend these theorems by showing that dynamic graphs of bounded (local) treewidth where the length of time over which the graph evolves and is observed is finite and bounded can be modelled in such a way that the (local) treewidth of the underlying graph is maintained. We show the application of these results to problems in dynamic graph theory and dynamic extensions to static problems. In addition we demonstrate that certain widely used dynamic graph classes naturally have bounded local treewidth.
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- 8.Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N.: Time-varying graphs and dynamic networks. IJPEDS 27(5), 387–408 (2012)Google Scholar
- 9.Chaintreau, A., Mtibaa, A., Massoulié, L., Diot, C.: The diameter of opportunistic mobile networks. In: CoNEXT, p. 12 (2007)Google Scholar
- 11.Cohen, R.F., Sairam, S., Tamassia, R., Vitter, J.S.: Dynamic algorithms for optimization problems in bounded tree-width graphs. In: Rinaldi, G., Wolsey, L.A. (eds.) Proceedings of the 3rd Integer Programming and Combinatorial Optimization Conference, Erice, Italy, pp. 99–112. CIACO (1993)Google Scholar
- 13.Courcelle, B.: The monadic second-order logic of graphs xvi: Canonical graph decompositions. Logical Methods in Computer Science 2(2) (2006)Google Scholar
- 14.Dvořák, Z., Král, D., Thomas, R.: Deciding first-order properties for sparse graphs. In: 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, Las Vegas, Nevada, pp. 133–142. IEEE Computer Society (2010)Google Scholar
- 17.Flum, J., Grohe, M.: Parameterized complexity theory. Springer (2006)Google Scholar
- 23.Kempe, D., Kleinberg, J.M., Kumar, A.: Connectivity and inference problems for temporal networks. In: STOC, pp. 504–513 (2000)Google Scholar
- 24.Kuhn, F., Lynch, N.A., Oshman, R.: Distributed computation in dynamic networks. In: STOC, pp. 513–522 (2010)Google Scholar