On the Treewidth of Dynamic Graphs

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


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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berman, K.A.: Vulnerability of scheduled networks and a generalization of Menger’s theorem. Networks 28, 125–134 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  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)MathSciNetzbMATHCrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  16. 16.
    Flocchini, P., Mans, B., Santoro, N.: On the exploration of time-varying networks. Theor. Comput. Sci. 469, 53–68 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  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)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Frick, M., Grohe, M.: Deciding first-order properties of locally tree-decomposable structures. Journal of the ACM 48(6), 1184–1206 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hagerup, T.: Dynamic algorithms for graphs of bounded treewidth. Algorithmica 27(3), 292–315 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  26. 26.
    Stewart, I.A.: On the fixed-parameter tractability of parameterized model-checking problems. Information Processing Letters 106(1), 33–36 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Bernard Mans
    • 1
  • Luke Mathieson
    • 1
  1. 1.Department of ComputingMacquarie UniversitySydneyAustralia

Personalised recommendations