Abstract
In this paper, we use the connection between dynamic monopolies and feedback vertex sets to establish explicit formulas and new bounds for decycling number \(\nabla (G)\) when \(G\) is one of the following classes of graphs: cycle permutation graphs, generalized Petersen graphs, and torus cordalis. In the first part of this paper, we show that if \(G\) is a cycle permutation graph or a generalized Petersen graph on \(2n\) vertices, then \(\nabla (G)=\lceil (n+1)/2\rceil \). These results extend a recent result by Zaker (Discret Math 312:1136–1143, 2012) and partially answer a question of Bau and Beineke (Australas J Comb 25:285–298, 2002). Note that our definition of a generalized Petersen graph is more general than the one used in Zaker (Discret Math 312:1136–1143, 2012). The second and major part of this paper is devoted to proving new upper bounds and exact values on the size of the minimum feedback vertex set and minimum dynamic monopoly for torus cordalis. Our results improve the previous results by Flocchini in 2004.
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Ackerman E, Ben-Zwi O, Wolfovitz G (2010) Combinatorial model and bounds for target set selection. Theoret Comput Sci 411:4017–4022
Adams SS, Troxell DS, Zinnen SL (2011) Dynamic monopolies and feedback vertex sets in hexagonal grids. Comput Math Appl 62:4049–4057
Albertson MO, Berman D (1979) A conjecture on planar graphs. Graph theory and related topics. Academic Press, New York, p 357
Bau S, Beineke LW (2002) The decycling number of graphs. Australas J Comb 25:285–298
Beineke LW, Vandell RC (1997) Decycling graphs. J Graph Theory 25:59–77
Ben-Zwi O, Hermelin D, Lokshtanov D, Newman I (2011) Treewidth governs the complexity of target set selection. Discret Optim 8:87–96
Berger E (2001) Dynamic monopolies of constant size. J Comb Theory Ser B 83:191–200
Chartrand G, Harary F (1967) Planar permutation graphs. Ann Inst Henri Poincaré Sect B 3:433–438
Chartrand G, Frechen JB (1968) On the chromatic number of permutation graphs. In: Harary F (ed) Proof techniques in graph theory, proceedings of the second ann arbor graph theory conference. Academic Press, New York, pp 21–24
Chen N (2009) On the approximability of influence in social networks. SIAM J Discret Math 23:1400–1415
Chiang C-Y, Huang L-H, Li B-J, Wu J, Yeh H-G (2013) Some results on the target set selection problem. J Comb Optim 25:702–715
Chiang C-Y, Huang L-H, Yeh H-G (2013) Target set selection problem for honeycomb networks. SIAM J Discret Math 23:310–328
Coxeter HSM (1950) Self-dual configurations and regular graphs. Bull Am Math Soc 56:413–455
Domingos P, Richardson M (2001) Mining the network value of customers. In: Proceedings of ACM SIGKDD 2001, San Francisco, pp 57–66
Dreyer PA, Roberts FS (2009) Irreversible \(k\)-threshold processes: graph-theoretical threshold models of the spread of disease and of opinion. Discret Appl Math 157:1615–1627
Festa P, Pardalos PM, Resende MGC (2000) Feedback set problems. In: Du D-Z, Pardalos PM (eds) Handbook of combinatorial optimization, vol A. Kluwer Academic Publishers, Dordrecht, pp 209–259
Flocchini P, Geurts F, Santoro N (2001) Optimal irreversible dynamos in chordal rings. Discret Appl Math 113:23–42
Flocchini P, Královič R, Ruźička P, Roncato A, Santoro N (2003) On time versus size for monotone dynamic monopolies in regular topologies. J Discret Algorithms 1:129–150
Flocchini P, Lodi E, Luccio F, Pagli L, Santoro N (2004) Dynamic monopolies in tori. Discret Appl Math 137:197–212
Flocchini P (2009) Contamination and decontamination in majority-based systems. J Cell Autom 4:183–200
Focardi R, Luccio FL, Peleg D (2000) Feedback vertex set in hypercubes. Inform Process Lett 76:1–5
Garey MR, Johnson DS (1979) Computers and Intractability: a guide to the theory of NP-completeness. In: Freeman WH (ed) A series of books in the mathematical sciences. Freeman&Co, San Francisco
Kempe D, Kleinberg J, Tardos E (2003) Maximizing the spread of influence through a social network. In: Proceedings of the 9th ACM SIGKDD international conference on knowledge discovery and data mining, pp 137–146
Kempe D, Kleinberg J, Tardos E (2005) Influential nodes in a diffusion model for social networks. In: Proceedings of the 32th international colloquium on automata, languages and programming, pp 1127–1138
Li D, Liu Y (1999) A polynomial algorithm for finding the minimum feedback vertex set of a \(3\)-regular simple graph. Acta Math Sci 19:375–381
Luccio F (1998) Almost exact minimum feedback vertex set in meshes and butterflies. Inf Process Lett 66:59–64
Luccio F, Pagli L, Sanossian H (1999) Irreversible dynamos in butterflies. In: 6th international colloquium on structural information and communication complexity (SIROCCO), pp 204–218
Peleg D (1998) Size bounds for dynamic monopolies. Discret Appl Math 86:263–273
Peleg D (2002) Local majorities, coalitions and monopolies in graphs: a review. Theoret Comput Sci 282:231–257
Pike DA, Zou Y (2005) Decycling cartesian products of two cycles. SIAM J Discret Math 19:651–663
Richardson M, Domingos P (2002) Mining knowledge-sharing sites for viral marketing. In: Proceedings of the 8th ACM SIGKDD, Edmonton, Canada, pp 61–70
Roberts FS (2006) Graph-theoretical problems arising from defending against bioterrorism and controlling the spread of fires. In: Proceedings of the DIMACS/DIMATIA/Renyi combinatorial challenges conference, Piscataway, NJ
Roberts FS (2003) Challenges for discrete mathematics and theoretical computer science in the defense against bioterrorism. In: Banks HT, Castillo-Chave C (eds) Bioterrorism: mathematical modeling applications in homeland security, Front. Appl. Mathem., vol 28. SIAM, Philadelphia, pp 1–34
Shamir A (1979) A linear time algorithm for finding minimum cutsets in reduced graphs. SIAM J Comput 8:654–655
Speckenmeyer E (1988) On the feedback vertex sets and nonseparating independent sets in cubic graphs. J Graph Theory 12:405–412
Stueckle S, Ringeisen RD (1984) Generalized Petersen graphs which are cycle permutation graphs. J Comb Theory B 37:142–150
Ueno S, Kajitani Y, Gotoh S (1988) On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three. Discret Math 72:355–360
Wang C, Lloyd E, Soffa M (1985) Feedback vertex sets and cyclically reducible graphs. J ACM 32:296–313
Watkins ME (1969) A theorem on Tait colorings with an application to the generalized Petersen graphs. J Comb Theory 6:152–164
Yehuda B, Geiger D, Naor J, Roth RM (1994) Approximation algorithms for the vertex feedback set problem with applications to constraint satisfaction and Bayesian inference. In: Proceedings of the 5th annual ACM-SIAM symposium on discrete algorithms, pp 344–354
Zaker M (2012) On dynamic monopolies of graphs with general thresholds. Discret Math 312:1136–1143
Acknowledgments
C.-Y. Chiang partially supported by NSC under Grant NSC97-2628-M-008-018-MY3. W.-T. Huang partially supported by NSC under Grant NSC100-2115-M-008-007-MY2. H.-G. Yeh partially supported by NSC under Grant NSC102-2115-M-008-011-MY2. A preliminary version of this paper can be downloaded from arXiv:1112.1313v1.
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Chiang, CY., Huang, WT. & Yeh, HG. Dynamic monopolies and feedback vertex sets in cycle permutation graphs, generalized Petersen graphs and torus cordalis. J Comb Optim 31, 815–832 (2016). https://doi.org/10.1007/s10878-014-9790-5
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DOI: https://doi.org/10.1007/s10878-014-9790-5