COCOON 2013: Computing and Combinatorics pp 349-360

# On the Treewidth of Dynamic Graphs

• Bernard Mans
• Luke Mathieson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7936)

## Abstract

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.

## Keywords

Static Graph Order Logic Tree Decomposition Sparse Graph Dynamic Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Berman, K.A.: Vulnerability of scheduled networks and a generalization of Menger’s theorem. Networks 28, 125–134 (1996)
2. 2.
Bhadra, S., Ferreira, A.: Complexity of connected components in evolving graphs and the computation of multicast trees in dynamic networks. In: Pierre, S., Barbeau, M., Kranakis, E. (eds.) ADHOC-NOW 2003. LNCS, vol. 2865, pp. 259–270. Springer, Heidelberg (2003)
3. 3.
Bodlaender, H.L.: Dynamic algorithms for graphs with treewidth 2. In: van Leeuwen, J. (ed.) WG 1993. LNCS, vol. 790, pp. 112–124. Springer, Heidelberg (1994)
4. 4.
Brejová, B., Dobrev, S., Královič, R., Vinař, T.: Routing in carrier-based mobile networks. In: Kosowski, A., Yamashita, M. (eds.) SIROCCO 2011. LNCS, vol. 6796, pp. 222–233. Springer, Heidelberg (2011)
5. 5.
Bui-Xuan, B.-M., Ferreira, A., Jarry, A.: Computing shortest, fastest, and foremost journeys in dynamic networks. Int. J. Found. Comput. Sci. 14(2), 267–285 (2003)
6. 6.
Casteigts, A., Chaumette, S., Ferreira, A.: Characterizing topological assumptions of distributed algorithms in dynamic networks. In: Kutten, S., Žerovnik, J. (eds.) SIROCCO 2009. LNCS, vol. 5869, pp. 126–140. Springer, Heidelberg (2010)
7. 7.
Casteigts, A., Flocchini, P., Mans, B., Santoro, N.: Deterministic computations in time-varying graphs: Broadcasting under unstructured mobility. In: Calude, C.S., Sassone, V. (eds.) TCS 2010. IFIP AICT, vol. 323, pp. 111–124. Springer, Heidelberg (2010)
8. 8.
Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N.: Time-varying graphs and dynamic networks. IJPEDS 27(5), 387–408 (2012)Google Scholar
9. 9.
Chaintreau, A., Mtibaa, A., Massoulié, L., Diot, C.: The diameter of opportunistic mobile networks. In: CoNEXT, p. 12 (2007)Google Scholar
10. 10.
Clementi, A.E.F., Monti, A., Silvestri, R.: Modelling mobility: A discrete revolution. Ad Hoc Networks 9(6), 998–1014 (2011)
11. 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
12. 12.
Courcelle, B.: The monadic second-order logic of graphs. i. recognizable sets of finite graphs. Information and Computation 85(1), 12–75 (1990)
13. 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. 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
15. 15.
Ferreira, A.: Building a reference combinatorial model for manets. IEEE Network 18(5), 24–29 (2004)
16. 16.
Flocchini, P., Mans, B., Santoro, N.: On the exploration of time-varying networks. Theor. Comput. Sci. 469, 53–68 (2013)
17. 17.
Flum, J., Grohe, M.: Parameterized complexity theory. Springer (2006)Google Scholar
18. 18.
Frederickson, G.N.: Maintaining regular properties dynamically in k-terminal graphs. Algorithmica 22(3), 330–350 (1998)
19. 19.
Frick, M., Grohe, M.: Deciding first-order properties of locally tree-decomposable structures. Journal of the ACM 48(6), 1184–1206 (2001)
20. 20.
Hagerup, T.: Dynamic algorithms for graphs of bounded treewidth. Algorithmica 27(3), 292–315 (2000)
21. 21.
Jacquet, P., Mans, B., Mühlethaler, P., Rodolakis, G.: Opportunistic routing in wireless ad hoc networks: Upper bounds for the packet propagation speed. IEEE Journal on Selected Areas in Communications 27(7), 1192–1202 (2009)
22. 22.
Jacquet, P., Mans, B., Rodolakis, G.: Information Propagation Speed in Mobile and Delay Tolerant Networks. IEEE Transactions on Information Theory 56(1), 5001–5015 (2010)
23. 23.
Kempe, D., Kleinberg, J.M., Kumar, A.: Connectivity and inference problems for temporal networks. In: STOC, pp. 504–513 (2000)Google Scholar
24. 24.
Kuhn, F., Lynch, N.A., Oshman, R.: Distributed computation in dynamic networks. In: STOC, pp. 513–522 (2010)Google Scholar
25. 25.
Kuhn, F., Oshman, R.: Dynamic networks: models and algorithms. SIGACT News 42(1), 82–96 (2011)
26. 26.
Stewart, I.A.: On the fixed-parameter tractability of parameterized model-checking problems. Information Processing Letters 106(1), 33–36 (2008)