Zero forcing propagation time on oriented graphs

Abstract Zero forcing is an iterative coloring procedure on a graph that starts by initially coloring vertices white and blue and then repeatedly applies the following rule: if any blue vertex has a unique (out-)neighbor that is colored white, then that neighbor is forced to change color from white to blue. An initial set of blue vertices that can force the entire graph to blue is called a zero forcing set. In this paper we consider the minimum number of iterations needed for this color change rule to color all of the vertices blue, also known as the propagation time, for oriented graphs. We produce oriented graphs with both high and low propagation times, consider the possible propagation times for the orientations of a fixed graph, and look at balancing the size of a zero forcing set and the propagation time.

[1]  Dieter Rautenbach,et al.  Extremal values and bounds for the zero forcing number , 2016, Discret. Appl. Math..

[2]  Shaun M. Fallat,et al.  On the minimum rank of not necessarily symmetric matrices : a preliminary study , 2009 .

[3]  Yair Caro,et al.  Upper bounds on the k-forcing number of a graph , 2014, Discret. Appl. Math..

[4]  Michael Young,et al.  Throttling zero force propagation speed on graphs , 2013, Australas. J Comb..

[5]  Simone Severini,et al.  Zero Forcing, Linear and Quantum Controllability for Systems Evolving on Networks , 2011, IEEE Transactions on Automatic Control.

[6]  Vittorio Giovannetti,et al.  Full control by locally induced relaxation. , 2007, Physical review letters.

[7]  Jean-Charles Delvenne,et al.  Zero forcing number, constraint matchings and strong structural controllability , 2014, ArXiv.

[8]  Ashkan Aazami,et al.  Hardness results and approximation algorithms for some problems on graphs , 2008 .

[9]  Nathaniel Dean,et al.  Iteration Index of a Zero Forcing Set in a Graph , 2011, 1105.1492.

[10]  Shaun M. Fallat,et al.  Minimum Rank, Maximum Nullity, and Zero Forcing Number of Graphs , 2014 .

[11]  L. Hogben.  Minimum Rank Problems , 2010 .

[12]  Robin J. Wilson,et al.  An Atlas of Graphs , 1999 .

[13]  Michael Young,et al.  Path cover number, maximum nullity, and zero forcing number of oriented graphs and other simple digraphs , 2015 .

[14]  Baoyindureng Wu,et al.  Proof of a conjecture on the zero forcing number of a graph , 2016, Discret. Appl. Math..

[15]  David E. Roberson,et al.  Fractional zero forcing via three-color forcing games , 2016, Discret. Appl. Math..

[16]  W. Haemers.  Zero forcing sets and minimum rank of graphs , 2008 .

[17]  Michael Young,et al.  MINIMUM RANK, MAXIMUM NULLITY, AND ZERO FORCING NUMBER OF SIMPLE DIGRAPHS , 2013 .

[18]  Sarah Meyer,et al.  Propagation time for zero forcing on a graph , 2012, Discret. Appl. Math..