Advertisement

On Graphs with Minimal Eternal Vertex Cover Number

  • Jasine BabuEmail author
  • L. Sunil Chandran
  • Mathew Francis
  • Veena Prabhakaran
  • Deepak Rajendraprasad
  • J. Nandini Warrier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11394)

Abstract

The eternal vertex cover problem is a variant of the classical vertex cover problem where a set of guards on the vertices have to be dynamically reconfigured from one vertex cover to another in every round of an attacker-defender game. The minimum number of guards required to protect a graph from an infinite sequence of attacks is the eternal vertex cover number (evc) of the graph. It is known that, given a graph G and an integer k, checking whether \({\text {evc}}(G) \le k\) is NP-Hard. However, for any graph G, \({\text {mvc}}(G) \le {\text {evc}}(G) \le 2 {\text {mvc}}(G)\), where \({\text {mvc}}(G)\) is the minimum vertex cover number of G. Precise value of eternal vertex cover number is known only for certain very basic graph classes like trees, cycles and grids. Though a characterization is known for graphs for which \({\text {evc}}(G) = 2{\text {mvc}}(G)\), a characterization of graphs for which \({\text {evc}}(G) = {\text {mvc}}(G)\) remained open. Here, we achieve such a characterization for a class of graphs that includes chordal graphs and internally triangulated planar graphs. For some graph classes including biconnected chordal graphs, our characterization leads to a polynomial time algorithm to precisely determine \({\text {evc}}(G)\) and to determine a safe strategy of guard movement in each round of the game with \({\text {evc}}(G)\) guards.

Keywords

Eternal vertex cover Chordal graphs Connected vertex cover 

References

  1. 1.
    Klostermeyer, W., Mynhardt, C.: Edge protection in graphs. Australas. J. Comb. 45, 235–250 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Fomin, F.V., Gaspers, S., Golovach, P.A., Kratsch, D., Saurabh, S.: Parameterized algorithm for eternal vertex cover. Inf. Process. Lett. 110(16), 702–706 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anderson, M., Carrington, J.R., Brigham, R.C., Dutton, D.R., Vitray, R.P.: Graphs simultaneously achieving three vertex cover numbers. J. Comb. Math. Comb. Comput. 91, 275–290 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Klostermeyer, W.F., Mynhardt, C.M.: Graphs with equal eternal vertex cover and eternal domination numbers. Discret. Math. 311, 1371–1379 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Diestel, R.: Graph Theory. GTM, vol. 173. Springer, Heidelberg (2017).  https://doi.org/10.1007/978-3-662-53622-3CrossRefzbMATHGoogle Scholar
  6. 6.
    Chartrand, G., Pippert, R.E.: Locally connected graphs. Časopis pro pěstování matematiky 99(2), 158–163 (1974)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Vanderjagt, D.W.: Sufficient conditions for locally connected graphs. Časopis pro pěstování matematiky 99(4), 400–404 (1974)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Erdös, P., Palmer, E.M., Robinson, R.W.: Local connectivity of a random graph. J. Graph Theor. 7(4), 411–417 (1983)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hopcroft, J.E., Karp, R.M.: An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J. Comput. 1, 180–187 (1972)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jasine Babu
    • 1
    Email author
  • L. Sunil Chandran
    • 2
  • Mathew Francis
    • 3
  • Veena Prabhakaran
    • 1
  • Deepak Rajendraprasad
    • 1
  • J. Nandini Warrier
    • 4
  1. 1.Indian Institute of Technology PalakkadPalakkadIndia
  2. 2.Indian Institute of ScienceBangaloreIndia
  3. 3.Indian Statistical InstituteChennaiIndia
  4. 4.National Institute of Technology CalicutCalicutIndia

Personalised recommendations