The quest for optimal/stable paths in graphs concerns a few practical or theoretical areas. Taking part in the quest, this paper adopts an abstract, general, equilibrium-oriented approach: it uses (quasi-arbitrary) arc-labelled digraphs, and assumes little about the structure of the sought paths and the definition of equilibrium, i.e. optimality/stability. The paper gives both a sufficient condition and a necessary condition for equilibrium existence for every “graph”, pinpoints the difference between these conditions, and shows coincidence when optimality relates to a total order. These results are applied to network routing.


Labelled directed graph path preference equilibrium optimisation strict weak order sufficient condition necessary condition induction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fishburn, P.: Intransitive indifference in preference theory: A survey. Operations Research 18, 207–228 (1970)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Griffin, T., Sobrinho, J.: Metarouting. In: SIGCOMM 2005: Proceedings of the 2005 conference on Applications, technologies, architectures, and protocols for computer communications, pp. 1–12. ACM Press, New York (2005)CrossRefGoogle Scholar
  3. 3.
    Lawler, E.: Combinatorial Optimization: Networks and Matroids. Dover (2001)Google Scholar
  4. 4.
    Le Roux, S.: Graphs and path equilibria. Research report, INRIA (2007),

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Stéphane Le Roux
    • 1
  1. 1.École normale supérieure de LyonUniversité de Lyon, LIP, CNRS, INRIA, UCBL 

Personalised recommendations