Abstract
Signed graphs have been introduced to enrich graph structures expressing relationships between persons or general social entities, introducing edge signs to reflect the nature of the relationship, e.g., friendship or enmity. Independently, offensive alliances have been defined and studied for undirected, unsigned graphs. We join both lines of research and define offensive alliances in signed graphs, hence considering the nature of relationships. Apart from some combinatorial results, mainly on k-balanced and k-anti-balanced signed graphs (a newly introduced family of signed graphs), we focus on the algorithmic complexity of finding smallest offensive alliances, looking at a number of parameterizations. While the parameter solution size leads to an FPT result for unsigned graphs, we obtain \(\textsf {W}[2]\)-completeness for the signed setting. We introduce new parameters for signed graphs, e.g., distance to weakly balanced signed graphs, that could be of independent interest. We show that these parameters yield FPT results. Here, we make use of the recently introduced parameter neighborhood diversity for signed graphs.
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References
Arrighi, E., Feng, Z., Fernau, H., Mann, K., Qi, X., Wolf, P.: Defensive alliances in signed networks. Technical report, Cornell University (2023)
Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Mach. Learn. 56(1–3), 89–113 (2004)
Cartwright, D., Harary, F.: Structural balance: a generalization of Heider’s theory. Psychol. Rev. 63(5), 277–293 (1956)
Cattanéo, D., Perdrix, S.: The parameterized complexity of domination-type problems and application to linear codes. In: Gopal, T.V., Agrawal, M., Li, A., Cooper, S.B. (eds.) TAMC 2014. LNCS, vol. 8402, pp. 86–103. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06089-7_7
Chellali, M.: Trees with equal global offensive \(k\)-alliance and \(k\)-domination numbers. Opuscula Mathematica 30, 249–254 (2010)
Chellali, M., Haynes, T.W., Randerath, B., Volkmann, L.: Bounds on the global offensive \(k\)-alliance number in graphs. Discussiones Mathematicae Graph Theory 29(3), 597–613 (2009)
Cygan, M., et al.: On problems as hard as CNF-SAT. ACM Trans. Algorithms 12(3), 41:1–41:24 (2016)
Cygan, M., et al.: Parameterized Algorithms. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-319-21275-3
Davis, J.A.: Clustering and structural balance in graphs. Human Relat. 20, 181–187 (1967)
Dourado, M.C., Faria, L., Pizaña, M.A., Rautenbach, D., Szwarcfiter, J.L.: On defensive alliances and strong global offensive alliances. Disc. Appl. Math. 163(Part 2), 136–141 (2014). https://doi.org/10.1016/j.dam.2013.06.029
Favaron, O.: Offensive alliances in graphs. Discussiones Mathematicae - Graph Theory 24, 263–275 (2002)
Fernau, H., Raible, D.: Alliances in graphs: a complexity-theoretic study. In: Leeuwen, J., et al. (eds.) SOFSEM 2007, Proceedings, vol. II, pp. 61–70. Institute of Computer Science ASCR, Prague (2007)
Fernau, H., Rodríguez-Velázquez, J.A., Sigarreta, S.M.: Offensive \(r\)-alliances in graphs. Disc. Appl. Math. 157, 177–182 (2009)
Gaikwad, A., Maity, S.: On structural parameterizations of the offensive alliance problem. In: Du, D.-Z., Du, D., Wu, C., Xu, D. (eds.) COCOA 2021. LNCS, vol. 13135, pp. 579–586. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92681-6_45
Gaikwad, A., Maity, S.: Offensive alliances in graphs. Technical Report. 2208.02992, Cornell University, ArXiv/CoRR (2022)
Gaikwad, A., Maity, S., Tripathi, S.K.: Parameterized complexity of defensive and offensive alliances in graphs. In: Goswami, D., Hoang, T.A. (eds.) ICDCIT 2021. LNCS, vol. 12582, pp. 175–187. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-65621-8_11
Harary, F.: On the notion of balance of a signed graph. Michigan Math. J. 2, 143–146 (1953–54)
Harutyunyan, A.: Global offensive alliances in graphs and random graphs. Disc. Appl. Math. 164, 522–526 (2014)
Harutyunyan, A., Legay, S.: Linear time algorithms for weighted offensive and powerful alliances in trees. Theore. Comput. Sci. 582, 17–26 (2015)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)
Johnson, D.S., Szegedy, M.: What are the least tractable instances of Max Independent Set? In: Tarjan, R.E., Warnow, T.J. (eds.) Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 927–928. ACM/SIAM (1999)
Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the exponential time hypothesis. EATCS Bull. 105, 41–72 (2011)
Mohar, B.: Face cover and the genus problem for apex graphs. J. Comb. Theory Ser. B 82, 102–117 (2001)
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Feng, Z., Fernau, H., Mann, K., Qi, X. (2024). Offensive Alliances in Signed Graphs. In: Chen, X., Li, B. (eds) Theory and Applications of Models of Computation. TAMC 2024. Lecture Notes in Computer Science, vol 14637. Springer, Singapore. https://doi.org/10.1007/978-981-97-2340-9_20
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