Algebraic and Dynamic Graph Algorithms

  • K. ErciyesEmail author
Part of the Texts in Computer Science book series (TCS)


Algebraic graph theory is the study of algebraic methods to solve graph problems. We review algebraic solutions to the main graph problems in the first part of this chapter. Many real-life networks are represented by dynamic graphs in which new vertices/edges may be inserted and some vertices/edges may be deleted as time progresses. We describe few dynamic graph problems that can be solved by dynamic graph algorithms, and finally we give a brief description of the methods used in dynamic algebraic graph algorithms, which are used for dynamic graphs using linear algebraic techniques.


  1. 1.
    Bapat RB (2014) Graphs and matrices (Universitext), 2nd edn. Springer, Berlin (Chapters 3 and 4)Google Scholar
  2. 2.
    Baswana S, Gupta M, Sen S (2011) Fully dynamic maximal matching in O(log n) update time. In: 52nd annual IEEE symposium on foundations of computer science FOCS 2011, pp 383–392Google Scholar
  3. 3.
    Bhattacharya S, Henzinger M, Nanongkai D (2017) Fully dynamic approximate maximum matching and minimum vertex cover in \(O(log^3n)\) worst case update time. In: 28th ACM SIAM symposium on discrete algorithms (SODA17), pp 470–489Google Scholar
  4. 5.
    Demetrescu C, Italiano GF (2004) A new approach to dynamic all pairs shortest paths. J Assoc Comput Mach (JACM) 51(6):968–992MathSciNetCrossRefGoogle Scholar
  5. 6.
    Demetrescu C, Italiano GF (2006) Fully dynamic all pairs shortest paths with real edge weights. J Comput Syst Sci 72(5):813–837MathSciNetCrossRefGoogle Scholar
  6. 7.
    Demetrescu C, Finocchi I, Italiano GF (2004) Dynamic graphs. Handbook of data structures and applications. Computer and information science series. Chapman and Hall/CRC, Boca Raton (Sect. 36)Google Scholar
  7. 8.
    Demetrescu C, Finocchi I, Italiano GF (2013) Dynamic graph algorithms, 2nd edn. Handbook of graph theory. Chapman and Hall/CRC, Boca Raton (Sect. 10-2)Google Scholar
  8. 9.
    Eppstein D, Galil Z, Italiano GF, Nissenzweig A (1997) Sparsification a technique for speeding up dynamic graph algorithms. J Assoc Comput Mach 44:669–696MathSciNetCrossRefGoogle Scholar
  9. 10.
    Fiedler M (1973) Algebraic connectivity of graphs. Czechoslovak Math J 23:298–305MathSciNetzbMATHGoogle Scholar
  10. 11.
    Frederickson GN (1985) Data structures for on-line updating of minimum spanning trees, with applications. SIAM J Comput 14(4):781–798MathSciNetCrossRefGoogle Scholar
  11. 12.
    Henzinger MR, King V (1999) Randomized fully dynamic graph algorithms with polylogarithmic time per operation. J ACM 46(4):502–516MathSciNetCrossRefGoogle Scholar
  12. 13.
    Holm J, de Lichtenberg K, Thorup M (2001) Poly-logarithmic deterministic fully dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J Assoc Comput Mach 48(4):723–760MathSciNetCrossRefGoogle Scholar
  13. 14.
    Kepner J, Gilbert J (eds) (2011) Graph algorithms in the language of linear algebra. SIAMzbMATHGoogle Scholar
  14. 15.
    Lovazs L (1979) On determinants, matchings, and random algorithms. In: Budach L (ed) Fundamentals of computing theory. Akademia-Verlag, BerlinGoogle Scholar
  15. 16.
    Lovazs L, Plummer M (1986) Matching theory. Academic Press, BudapestGoogle Scholar
  16. 18.
    Mucha M, Sankowski P (2006) Maximum matchings in planar graphs via Gaussian elimination. Algorithmica 45(1):3–20MathSciNetCrossRefGoogle Scholar
  17. 19.
    Neiman O, Solomon S (2013) Simple deterministic algorithms for fully dynamic maximal matching. In: Proceedings of the ACM symposium on theory of computing (STOC’13), pp 745–754Google Scholar
  18. 20.
    Onak K, Rubinfeld R (2010) Maintaining a large matching and a small vertex cover. In: Proceedings of the ACM symposium on theory of computing (STOC’10), pp 457–464Google Scholar
  19. 21.
    Rabin MO, Vazirani VV (1989) Maximum matchings in general graphs through randomization. J Algorithms 10(4):557–567MathSciNetCrossRefGoogle Scholar
  20. 22.
    Sankowski P (2005) Algebraic graph algorithms. Ph.D. thesis, Warsaw University, Faculty of Mathematics, Information and MechanicsGoogle Scholar
  21. 26.
    Solomon S (2016) Fully dynamic maximal matching in constant update time. In: Proceedings of FOCS, pp 325–334Google Scholar
  22. 27.
    Tarjan R (1975) Efficiency of a good but not linear set union algorithm. J ACM 22(2):215–225MathSciNetCrossRefGoogle Scholar
  23. 28.
    Thorup M (2000) Near-optimal fully-dynamic graph connectivity. In: Proceedings of the thirty-second annual ACM symposium on Theory of computing. ACM Press, pp 343–350Google Scholar
  24. 29.
    Tutte WT (1947) The factorization of linear graphs. J Lond Math Soc s1–22(2):107–111MathSciNetCrossRefGoogle Scholar
  25. 30.
    Zavlanos MM, Pappas GJ (2005) Controlling connectivity of dynamic graphs. In: Proceedings of the 44th IEEE conference on decision and control and European control conference, Seville, Spain, December 2005, pp 6388–6393Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.International Computer InstituteEge UniversityIzmirTurkey

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