The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph that cover all the vertices of the graph, while the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph needed to cover all the vertices in the graph. We show that for a block-cycle graph the zero forcing number equals the path cover number.We also give a purely graph theoretical proof that the positive zero forcing number of any outerplanar graphs equals the tree cover number of the graph. These ideas are then extended to the setting of k-trees, where the relationship between the positive zero forcing number and the tree cover number becomes more complex.
@article{bwmeta1.element.doi-10_2478_spma-2014-0004,
author = {Fatemeh Alinaghipour Taklimi and Shaun Fallat and Karen Meagher},
title = {On the Relationships between Zero Forcing Numbers and Certain Graph Coverings},
journal = {Special Matrices},
volume = {2},
year = {2014},
zbl = {1291.05126},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_spma-2014-0004}
}
Fatemeh Alinaghipour Taklimi; Shaun Fallat; Karen Meagher. On the Relationships between Zero Forcing Numbers and Certain Graph Coverings. Special Matrices, Tome 2 (2014) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_spma-2014-0004/
[1] AIM Minimum Rank - Special Graphs Work Group. Zero forcing sets and the minimum rank of graphs. Linear Algebra Appl., 428/7: 1628-1648, 2008.[WoS] | Zbl 1135.05035
[2] F. Alinaghipour Taklimi. Zero Forcing Sets for Graphs University of Regina, 2013. Ph.D. thesis.
[3] E. Almodovar, L. DeLoss, L. Hogben, K. Hogenson, K. Myrphy, T. Peters, C. Ramrez, C. Minimum rank, maximum nullity and zero forcing number, and spreads of these parameters for selected graph families. Involve, 3: 371-392, 2010. | Zbl 1203.05088
[4] F. Barioli, W. Barrett, S. Fallat, H. T. Hall, L. Hogben, B. Shader, P. van den Driessche, and H. van der Holst. Zero forcing parameters and minimum rank problems. Linear Algebra Appl., 433(2): 401-411, 2010.[WoS] | Zbl 1209.05139
[5] F. Barioli, W. Barrett, S. Fallat, H. T. Hall, L. Hogben, B. Shader, P. van den Driessche, H. van der Holst. Parameters related to tree-width, zero forcing, and maximum nullity of a graph. J. Graph Theory, 72: 146-177, 2013.[WoS] | Zbl 1259.05112
[6] F. Barioli, S. Fallat, and L. Hogben. On the difference between the maximum multiplicity and path cover number for treelike graphs. Linear Algebra Appl., 409: 13-31, 2005. | Zbl 1072.05037
[7] F. Barioli, S. M. Fallat, L. H. Mitchell, and S. K. Narayan. Minimum semideffnite rank of outerplanar graphs and the tree cover number. Electron J. Linear Algebra, 22: 10-21, 2011. | Zbl 1205.05135
[8] D. Burgarth and V. Giovannetti. Full control by locally induced relaxation. Phys. Rev. Lett., 99(10): 100-501, 2007.
[9] D. Burgarth and K. Maruyama. Indirect Hamiltonian identiffcation through a small gateway. New J. Physics, 11(10): 103019, 2009.
[10] D. Burgarth, D. D’Alessandro, L. Hogben, S. Severini, M. Young. Zero forcing, linear and quantum controllability for systems evolving on networks. IEEE Transactions in Automatic Control, 58(9): 2349-2354, 2013.
[11] R. Diestal. Graph Theory. Graduate Texts in Mathematics, 137, Springer-Verlag, Heidelberg, 2000.
[12] C. J. Edholm, L. Hogben, M. Huynh, J. Lagrange, D. D. Row. Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph. Linear Algebra Appl., 436: 4352-4372, 2012.[WoS] | Zbl 1241.05076
[13] J. Ekstrand, C. Erickson, H. T. Hall, D. Hay, L. Hogben, R. Johnson, N. Kingsley, S. Osborne, T. Peters, J. Roat, et al. Positive semideffnite zero forcing. Linear Algebra Appl., 439: 1862-1874, 2013. | Zbl 1283.05165
[14] J. Ekstrand, C. Erickson, D. Hay, L. Hogben, and J. Roat. Note on positive semideffnite maximum nullity and positive semideffnite zero forcing number of partial 2-trees. Electron J. Linear Algebra, 23: 79-97, 2012. | Zbl 1252.05118
[15] P. Hackney, B. Harris, M. Lay, L. H. Mitchell, S. K. Narayan, and A. Pascoe. Linearly independent vertices and minimum semideffnite rank. Linear Algebra Appl., 431: 1105 - 1115, 2009.[WoS] | Zbl 1188.05085
[16] L.-H. Huang, G. J. Chang, H.-G. Yeh. On minimum rank and zero forcing sets of a graph. Linear Algebra Appl., 432: 2961-2973, 2010.[WoS] | Zbl 1195.05043
[17] AIM Minimum Rank-Special Graphs Work Group. Zero forcing sets and the minimum rank of graphs. Linear Algebra Appl., 428(7): 1628-1648, 2008.[WoS] | Zbl 1135.05035
[18] C. R. Johnson, R. Loewy, and P. A. Smith. The graphs for which the maximum multiplicity of an eigenvalue is two. Linear Multilinear Algebra, 57(7): 713-736, 2009.[Crossref][WoS] | Zbl 1225.05167
[19] D. Row. A technique for computing the zero forcing number of a graph with a cut-vertex. Linear Algebra Appl., 436(12): 4423-4432, 2012.[WoS] | Zbl 1241.05086
[20] D. Row. Zero forcing number, path cover number, and maximum nullity of cacti. Involve, 4(3): 277-291, 2011. | Zbl 1246.05098
[21] S. Severini. Nondiscriminatory propagation on trees. J. of Phys. A, 41: 482-002 (Fast Track Communication), 2008. | Zbl 1156.81357
[22] B. Yang. Fast-mixed searching and related problems on graphs. Theoret. Comput. Sci., 507(7): 100-113, 2013 | Zbl 1302.05197