On the Relationships between Zero Forcing Numbers and Certain Graph Coverings

• Autorzy: Fatemeh Alinaghipour Taklimi , Shaun Fallat , Karen Meagher
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Zero forcing number; positive zero forcing number; path cover number; tree cover number

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2014
• Tom: 2
• Numer: 1

Daty

wydano

2014-01-01

otrzymano

2013-10-25

zaakceptowano

2014-03-12

online

2014-04-24

Twórcy

autor

Fatemeh Alinaghipour Taklimi

• Department of Mathematics and Statistics, University of Regina, 3737 Wascana Parkway, S4S 0A4 Regina SK, Canada

autor

Shaun Fallat

• Department of Mathematics and Statistics, University of Regina, 3737 Wascana Parkway, S4S 0A4 Regina SK, Canada, Research supported by an NSERC Discovery Research Grant.

autor

Karen Meagher

• Department of Mathematics and Statistics, University of Regina, 3737 Wascana Parkway, S4S 0A4 Regina SK, Canada, Research supported by an NSERC Discovery Research Grant.

Bibliografia

• [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]
• [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.
• [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]
• [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]
• [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.
• [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.
• [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]
• [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.
• [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.
• [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]
• [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]
• [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]
• [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]
• [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]
• [20] D. Row. Zero forcing number, path cover number, and maximum nullity of cacti. Involve, 4(3): 277-291, 2011.
• [21] S. Severini. Nondiscriminatory propagation on trees. J. of Phys. A, 41: 482-002 (Fast Track Communication), 2008.
• [22] B. Yang. Fast-mixed searching and related problems on graphs. Theoret. Comput. Sci., 507(7): 100-113, 2013

Identyfikatory

DOI

10.2478/spma-2014-0004