Abstract
Target Set Selection (TSS) was initially proposed to study the problem of the spread of information, ideas or influence through a social network and had formulated many problems arising in various practical applications. We consider a particular type of graphs, namely n-dimensional m-sided pancake graph mPn, which is one class of Cayley graphs and is widely used in the symmetric interconnection networks. We establish a bound of TSS on mPn by the minimum feedback vertex set.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Ackerman, O. Ben-Zwi, and G. Wolfovitz, Combinatorial model and bounds for target set selection, Theoret. Comput. Sci., 411 (2010), 4017–4022.
S. S. Adams, Z. Brass, C. Stokes, and D. S. Troxell, Irreversible k-threshold and majority conversion processes on complete multipartite graphs and graph products, Australas. J. Combin., 56 (2013), 47–60.
S. S. Adams, D. S. Troxell, and S. L. Zinnen, Dynamic monopolies and feedback vertex sets in hexagonal grids, Comput. Math. Appl., 62 (2011), 4049–4057.
S. Bao and L. W. Beineke, The decycling number of graphs, Australas. J. Combin., 25 (2002), 285–298.
O. Ben-Zwi, D. Hermelin, and D. Lokshtanov, An exact almost optimal algorithm for target set selection in social networks, Proceedings of the tenth ACM conference on Electronic commerce, 2009, Stanford, CA, USA.
L. Beineke and R. Vandell, Decycling graphs, J. Graph Theory, 25 (1997), 59–77.
E. Berger, Dynamic monopolies of constant size, J. Combin. Theory Ser. B, 83 (2001), 191–200.
N. Chen, On the approximability of influence in social networks, SIAM J. Discrete Math., 23 (2009), 1400–1415.
C. Y. Chiang, L. H. Huang, and H. G. Yeh. Target set selection problem for honeycomb networks, SIAM J. Discrete Math., 27 (2013), 310–328.
C. Y. Chiang, L. H. Huang, B. J. Li, J. Wu, and H. G. Yeh, Some results on the target set selection problem, J. Comb. Optim., 25 (2011), 702–715.
M. Chopin, A. Nichterlein, R. Niedermeier, and M. Weller, Constant thresholds can make target set selection tractable, Theory. Comput. Systems, 55 (2014), 61–83.
P. Compeau, Girth of pancake graphs, Discrete Appl. Math., 159 (2011), 1641–1645.
P. Domingos and M. Richardson, Mining the network value of customers, Seventh International Conference on Knowledge Discovery and Data Mining, 2001, San Francisco, CA, USA.
P. A. Dreyer and F. S. Roberts, Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion, Discrete Appl. Math., 157 (2009), 1615–1627.
T. Iwasaki, Fault-tolerant routing in burnt pancake graphs, Inform. Process. Lett., 110 (2010), 535–538.
P. Festa, P. Pardalos, and Mauricio G. C. Resende, Feedback set problems, Springer, US, (1999).
P. Flocchini, R. Královič, P. Ruzicka, and N. Santoro, On time versus size for monotone dynamic monopolies in regular topologies, J. Discrete Algorithms, 2 (2002), 135–156.
P. Flocchini, E. Lodi, F. Luccio, L. Pagli, and N. Santoro, Dynamic monopolies in tori, Discrete Appl. Math., 137 (2004), 197–212.
M. Heydari and I. Sudborough, On sorting by prefix reversals and the dimaeter of pancake networks, Parallel Archirectures and Their Efficient Use, Lecture Notes in Computer Science, 678 (1993), 218–227.
C. Hoffman, Group theoretic algorithms and graph isomorphism, Springer, New York (1982).
S. Jung and K. Kaneko, A feedback vertex set on pancake graphs, International Conference on Parallel and distributed Processing Techniques and Applications, 2010, San Francisco, CA, USA.
M. Justan, F. Muga II, and I. Sudborough, On the generalization of the pancake network, Proceedings of the International Symposium on Parallel Archiectures, Algorithms and Networks, 2002, Sydney, Australia.
K. Kaneko and Y. Suzuki, Node-to-set disjoint paths problem in pancake graphs, IEICE Trans. Inf. Syst., E86-D (2003), 1628–1633.
D. Kempe, J. Kleinberg, and E. Tardos, Maximizing the spread of influence through a social network, In Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2003, London, UK.
F. Luccio, Almost exact minimum feedback vertex set in meshes and butterflies, Inform. Process. Lett., 66 (1998), 59–64.
D. Peleg, Local majorities, coalitions and monopolies in graphs: A review, Theoret. Comput. Sci., 282 (2002), 231–257.
D. A. Pike and Y. Zou, Decycling Cartesian products of two cycles, SIAM J. Discrete Math., 19 (2005), 651–663.
M. Zaker, On dynamic monopolies of graphs with general thresholds, Discrete Math., 312 (2012), 1136–1143.
Acknowledgement
My deepest gratitude goes to the anonymous reviewers for their careful work and thoughtful suggestions that have helped improve this paper substantially.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the Doctoral Foundation of Heze University (Grant numbers, XY18BS12).
Rights and permissions
About this article
Cite this article
Jiang, H. Target Set Selection on generalized pancake graphs. Indian J Pure Appl Math 51, 379–389 (2020). https://doi.org/10.1007/s13226-020-0406-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-020-0406-8