Zero forcing in Benzenoid network
DOI:
https://doi.org/10.22199/issn.0717-6279-2019-05-0064Keywords:
Zero forcing set, Pyrene networks, Circum-pyrene networks, Circum-trizene networksAbstract
A set S of vertices in a graph G is called a dominating set of G if every vertex in V (G)\S is adjacent to some vertex in S. A set S is said to be a power dominating set of G if every vertex in the system is monitored by the set S following a set of rules for power system monitoring. The power domination number of G is the minimum cardinality of a power dominating set of G. A dynamic coloring of the vertices of a graph G starts with an initial subset S of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set S is called a forcing set (zero forcing set) of G if, by iteratively applying the forcing process, every vertex in G becomes colored. The zero forcing number of G, denoted Z(G), is the minimum cardinality of a zero forcing set of G. In this paper, we obtain the zero forcing number for certain benzenoid networks.
References
AIM Minimum Rank – Special Graphs Work Group, “Zero forcing sets and the minimum rank of graphs”, Linear algebra and its applications, vol. 428, no. 7, pp. 1628–1648, Apr. 2008, doi: 10.1016/j.laa.2007.10.009 .
D. Burgarth, V. Giovannetti, L. Hogben, S. Severini and M. Young, “Logic circuits from zero forcing”, Natural computing, vol. 14, no. 3, pp. 485-490, Sep. 2015, doi: 10.1007/s11047-014-9438-5.
M. Gentner, L. Penso, D. Rautenbach and U. Souza, “Extremal values and bounds for the zero forcing number”, Discrete applied aathematics, vol. 214, pp. 196-200, Dec. 2016, doi: 10.1016/j.dam.2016.06.004.
T. Haynes, S. Hedetniemi, S. Hedetniemi, and M. Henning, “Power domination in graphs applied to electrical power networks”. SIAM journal on discrete mathematics, vol. 15, no. 4, pp. 519-529, 2002, doi: 10.1137/S0895480100375831.
K. Benson, D. Ferrero, M. Flagg, V. Furst, L. Hogben, V. Vasilevska and B. Wissman, “Zero forcing and power domination for product graphs”, Feb. 2017. arXiv:1510.02421v4.
J. Anitha and I. Rajasingh “Power domination parameters in honeycomb-like networks,” in Applied mathematics and scientific computing. Trends in mathematics. B. Rushi, R. Sivaraj, B. Prasad, M. Nalliah, and A. Reddy Eds. Cham: Birkhäuser, 2019, pp. 613–621, doi: 10.1007/978-3-030-01123-9_61.
J. Quadras, K. Balasubramanian and K. Arputha Christy, “Analytical expressions for Wiener indices of n-circumscribed pericondensed benzenoid graphs”, Journal of mathematical chemistry, vol. 54. no. 3, Mar. 2016, doi: 10.1007/s10910-016-0596-9.
_(1).png)
