K-Forcing Number of Some Graphs and Their Splitting Graphs

remodkmuar K P¹ , Charles Dominic² , Baby Chacko³

Section:Research Paper, Product Type: Isroset-Journal
Vol.6 , Issue.3 , pp.121-127, Jun-2019

CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i3.121127

Online published on Jun 30, 2019

Copyright © Premodkmuar K P , Charles Dominic , Baby Chacko . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
Amos et al. instigated the idea of k-forcing number of a graph. For a graph G=(V,E) and a subset Fof G, the vertices in F are called initially colored black vertices and the vertices in V(G)-F are called initially non colored black vertices or white vertices. Then the set F is a k-forcing set of a graph G if all vertices in G will become colored black after applying the subsequent color changing rule. Color changing rule: If a black colored vertex is adjacent to atmost k-white vertices, then the white vertices change to be colored black. The cardinality of a smallest k-forcing set is known as the k-forcing number of the graph G and is represented as Z_k (G). This work is intended to investigate the k-forcing number of the splitting graph of a graph in which k≥2.

Key-Words / Index Term :
Zero Forcing Number, K-forcing Number and Splitting Graph

References :
[1]. E. Sampathkumar and H.B. Walikaer, “On the splitting graph of a graph”, Journal of Karnatak University Science, Vol.25, pp.13-16, 1981.
[2]. K. P. Premodkumar, Ch. Dominic and B. Chacko, “On the Zero Forcing Number of Graphs and Their Splitting Graphs”, Algebra and Discrete Mathematics, Accepted, 2018.
[3]. H. Van der Holst et al., “Zero forcing sets and the minimum rank of graphs”, Linear Algebra and its Applications, Vol. 428, pp. 1628–1648, 2008.
[4]. D. Burgarth and V. Giovannetti. “Full control by locally induced relaxation”, Physical review letters, Vol. 99, Issue.10,pp 100501-100504, 2007.
[5]. D. Burgarth, V. Giovannetti, L. Hogben, S. Severini, and M. Young. “Logic circuits from zero forcing”.arXiv preprint arXiv:1106.4403, 2011.
[6]. L. Hogben, D. Burgarth, D. Alessandro, S.Severini, and M. Young. “Zero forcing, linear and quantum controllability for systems evolving on networks”, IEEE Transactions on Automatic Control, vol. 58, Issue. 9, pp. 2349-2354, 2013.
[7]. D Amos, Y caro, R Davila and R Pepper, “Upper bounds on the k-forcing number of a graph”, Discrete Applied Mathematics,Vol.181, pp 1–10, 2015.
[8]. E. Sampathkumar, M.S. Subramanya and Ch. Dominic, “3-Consecutive vertex Coloring of a Graph”, Proc. ICDM 2008, RMS-Lecture Notes Series Vol.13, pp.161-170, 2010.
[9]. H. Whitney, “Congruent graphs and the connectivity of graphs”. Amer. J. Math. Vol.54,pp 150-168, 1932.
[10]. F. Harary, Graph Theory, Addison-Wesely, Massachusethes, 1969.
[11]. R. Frucht and F. Harary, “On the Corona of two Graphs”, A equationes Math, Vol. 4 pp. 322-325, 1970.
[12]. M. Hladnik., Marusic, D. and Pisanski, “Cyclic Haar Graphs”, Discrete. Mathematics. Vol 244, pp.137-153, 2002.
[13]. Y Zhao, L Chen and H Li., “On Tight Bounds for the k-Forcing Number of a Graph”, Bull. Malays. Math. Sci. Sco., DOI 10.1007/s40840-017-0507-7.