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A connectivity game for graphs

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  • Rafael Amer
  • José Miguel Giménez

Abstract

The introduction of a {0,1}-valued game associated to a connected graph allows us to assign to each vertex a value of weighted connectivity according to the different solutions that for cooperative games are obtained by means of semivalues. The marginal contributions of each vertex to the coalitions differentiate an active connectivity from another reactive connectivity, according to whether the vertex is essential to obtain the connection or is the obstacle for the connection between the vertices in the coalition. We offer general properties of the connectivity, as well as the behaviour of different families of graphs with regard to this concept. We also analyse the effect on different vertices due to the addition of an edge to the initial graph. Copyright Springer-Verlag 2004

Suggested Citation

  • Rafael Amer & José Miguel Giménez, 2004. "A connectivity game for graphs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 60(3), pages 453-470, December.
    • Handle: RePEc:spr:mathme:v:60:y:2004:i:3:p:453-470
    DOI: 10.1007/s001860400356
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    Citations

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    Cited by:

    1. Ramón Flores & Elisenda Molina & Juan Tejada, 2019. "Evaluating groups with the generalized Shapley value," 4OR, Springer, vol. 17(2), pages 141-172, June.
    2. Molina, Elisenda & Tejada, Juan, 2013. "The Shapley group value," DES - Working Papers. Statistics and Econometrics. WS ws133430, Universidad Carlos III de Madrid. Departamento de Estadística.
    3. Giulia Cesari & Roberto Lucchetti & Stefano Moretti, 2017. "Generalized additive games," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(4), pages 919-939, November.
    4. Amer, Rafael & Gimenez, Jose Miguel & Magana, Antonio, 2007. "Accessibility in oriented networks," European Journal of Operational Research, Elsevier, vol. 180(2), pages 700-712, July.
    5. Lindelauf, R. & Hamers, H.J.M. & Husslage, B.G.M., 2011. "Game Theoretic Centrality Analysis of Terrorist Networks : The Cases of Jemaah Islamiyah and Al Qaeda," Discussion Paper 2011-107, Tilburg University, Center for Economic Research.
    6. Lindelauf, R. & Hamers, H.J.M. & Husslage, B.G.M., 2011. "Game Theoretic Centrality Analysis of Terrorist Networks : The Cases of Jemaah Islamiyah and Al Qaeda," Other publications TiSEM e91f6269-bcac-4790-b42f-3, Tilburg University, School of Economics and Management.
    7. Tom C. van der Zanden & Hans L. Bodlaender & Herbert J. M. Hamers, 2023. "Efficiently computing the Shapley value of connectivity games in low-treewidth graphs," Operational Research, Springer, vol. 23(1), pages 1-23, March.
    8. Husslage, B.G.M. & Lindelauf, R. & Hamers, H.J.M., 2012. "Leaderless Covert Networks : A Quantitative Approach," Other publications TiSEM c4111fba-0f6c-42cd-976a-1, Tilburg University, School of Economics and Management.
    9. Lindelauf, R.H.A. & Hamers, H.J.M. & Husslage, B.G.M., 2013. "Cooperative game theoretic centrality analysis of terrorist networks: The cases of Jemaah Islamiyah and Al Qaeda," European Journal of Operational Research, Elsevier, vol. 229(1), pages 230-238.
    10. Stefano Moretti & Henk Norde, 2022. "Some new results on generalized additive games," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(1), pages 87-118, March.
    11. Husslage, B.G.M. & Lindelauf, R. & Hamers, H.J.M., 2012. "Leaderless Covert Networks : A Quantitative Approach," Discussion Paper 2012-057, Tilburg University, Center for Economic Research.

    More about this item

    Keywords

    Cooperative game; Graph; Connectivity; Semivalue; 91A12; 91A43;
    All these keywords.

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