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Eternal Connected Vertex Cover Problem

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Theory and Applications of Models of Computation (TAMC 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12337))

Abstract

A new variant of the classic problem of Vertex Cover (VC) is introduced. Either Connected Vertex Cover (CVC) or Eternal Vertex Cover (EVC) is already a variant of VC, and Eternal Connected Vertex Cover (ECVC) is an eternal version of CVC, or a connected version of EVC. A connected vertex cover of a connected graph G is a vertex cover of G inducing a connected subgraph in G. CVC is the problem of computing a minimum connected vertex cover. For EVC a multi-round game on G is considered, and in response to every attack on some edge e of G, a guard positioned on a vertex of G must cross e to repel it. EVC asks the minimum number of guards to be placed on the vertices of a given G, that is sufficient to repel any sequence of edge attacks of an arbitrary length.

ECVC is EVC in which any configuration of guards, that is the set of vertices occupied by them, needs to be kept connected in each round. This paper presents, besides some basic structural properties of ECVC, 1) a polynomial time algorithm for ECVC on chordal graphs, 2) NP-completeness of ECVC on locally connected graphs, 3) a complete characterization of ECVC on cactus graphs, block graphs, or any graphs in which every block is either a simple cycle or a clique, and 4) a 2-approximation algorithm for ECVC on general graphs.

This work is supported in part by JSPS KAKENHI under Grant Number 17K00013.

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References

  1. Anderson, M., Barrientos, C., Brigham, R.C., Carrington, J.R., Vitray, R.P., Yellen, J.: Maximum-demand graphs for eternal security. J. Combin. Math. Combin. Comput. 61, 111–127 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Arkin, E.M., Halldórsson, M.M., Hassin, R.: Approximating the tree and tour covers of a graph. Inform. Process. Lett. 47(6), 275–282 (1993)

    Article  MathSciNet  Google Scholar 

  3. Babu, J., Chandran, L.S., Francis, M., Prabhakaran, V., Rajendraprasad, D., Warrier, J.N.: On graphs with minimal eternal vertex cover number. In: Pal, S.P., Vijayakumar, A. (eds.) CALDAM 2019. LNCS, vol. 11394, pp. 263–273. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11509-8_22

    Chapter  Google Scholar 

  4. Blažej, V., Křišt’an, J.M., Valla, T.: On the \(m\)-eternal domination number of cactus graphs. In: Filiot, E., Jungers, R., Potapov, I. (eds.) RP 2019. LNCS, vol. 11674, pp. 33–47. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-30806-3_4

    Chapter  Google Scholar 

  5. Burger, A.P., Cockayne, E.J., Gründlingh, W.R., Mynhardt, C.M., van Vuuren, J.H., Winterbach, W.: Infinite order domination in graphs. J. Combin. Math. Combin. Comput. 50, 179–194 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Cockayne, E.J., Dreyer Jr., P.A., Hedetniemi, S.M., Hedetniemi, S.T.: Roman domination in graphs. Discrete Math. 278(1–3), 11–22 (2004)

    Article  MathSciNet  Google Scholar 

  7. Diestel, R.: Graph Theory. GTM, vol. 173. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-53622-3

    Book  MATH  Google Scholar 

  8. Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann Math (2) 162(1), 439–485 (2005)

    Article  MathSciNet  Google Scholar 

  9. Escoffier, B., Gourvès, L., Monnot, J.: Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs. J. Discrete Algorithms 8(1), 36–49 (2010)

    Article  MathSciNet  Google Scholar 

  10. Fernau, H., Manlove, D.F.: Vertex and edge covers with clustering properties: complexity and algorithms. J. Discrete Algorithms 7(2), 149–167 (2009)

    Article  MathSciNet  Google Scholar 

  11. 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)

    Article  MathSciNet  Google Scholar 

  12. Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)

    Article  MathSciNet  Google Scholar 

  13. Goddard, W., Hedetniemi, S.M., Hedetniemi, S.T.: Eternal security in graphs. J. Combin. Math. Combin. Comput. 52, 169–180 (2005)

    MathSciNet  MATH  Google Scholar 

  14. Goldwasser, J.L., Klostermeyer, W.F.: Tight bounds for eternal dominating sets in graphs. Discrete Math. 308(12), 2589–2593 (2008)

    Article  MathSciNet  Google Scholar 

  15. Klostermeyer, W.F.: An eternal vertex cover problem. J. Combin. Math. Combin. Comput. 85, 79–95 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Klostermeyer, W.F., MacGillivray, G.: Eternal dominating sets in graphs. J. Combin. Math. Combin. Comput. 68, 97–111 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Klostermeyer, W.F., Mynhardt, C.M.: Edge protection in graphs. Australas. J. Combin. 45, 235–250 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Klostermeyer, W.F., Mynhardt, C.M.: Graphs with equal eternal vertex cover and eternal domination numbers. Discrete Math. 311(14), 1371–1379 (2011)

    Article  MathSciNet  Google Scholar 

  19. Klostermeyer, W.F., Mynhardt, C.M.: Eternal total domination in graphs. Ars Combin. 107, 473–492 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Li, Y., Wang, W., Yang, Z.: The connected vertex cover problem in \(k\)-regular graphs. J. Comb. Optim. 38(2), 635–645 (2019). https://doi.org/10.1007/s10878-019-00403-3

    Article  MathSciNet  MATH  Google Scholar 

  21. Priyadarsini, P.L.K., Hemalatha, T.: Connected vertex cover in 2-connected planar graph with maximum degree 4 is NP-complete. Int. J. Math. Phys. Eng. Sci. 2(1), 51–54 (2008)

    Google Scholar 

  22. Savage, C.: Depth-first search and the vertex cover problem. Inform. Process. Lett. 14(5), 233–235 (1982)

    Article  MathSciNet  Google Scholar 

  23. Ueno, S., Kajitani, Y., Gotoh, S.: On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three. Discrete Math. 72, 355–360 (1988)

    Article  MathSciNet  Google Scholar 

  24. Watanabe, T., Kajita, S., Onaga, K.: Vertex covers and connected vertex covers in 3-connected graphs. In: IEEE International Symposium on Circuits and Systems, pp. 1017–1020 (1991)

    Google Scholar 

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Authors and Affiliations

  1. Toyohashi University of Technology, Toyohashi, 441-8580, Japan

    Toshihiro Fujito & Tomoya Nakamura

Authors
  1. Toshihiro Fujito
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  2. Tomoya Nakamura
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Correspondence to Toshihiro Fujito .

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Editors and Affiliations

  1. Department of Computer Science, Texas A&M University, College Station, TX, USA

    Jianer Chen

  2. School of Computer Science and Engineering, Central South University, Changsha, China

    Qilong Feng

  3. Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, NY, USA

    Jinhui Xu

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Fujito, T., Nakamura, T. (2020). Eternal Connected Vertex Cover Problem. In: Chen, J., Feng, Q., Xu, J. (eds) Theory and Applications of Models of Computation. TAMC 2020. Lecture Notes in Computer Science(), vol 12337. Springer, Cham. https://doi.org/10.1007/978-3-030-59267-7_16

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  • DOI: https://doi.org/10.1007/978-3-030-59267-7_16

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-59266-0

  • Online ISBN: 978-3-030-59267-7

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