68,049 research outputs found
Fractional Zero Forcing via Three-color Forcing Games
An -fold analogue of the positive semidefinite zero forcing process that
is carried out on the -blowup of a graph is introduced and used to define
the fractional positive semidefinite forcing number. Properties of the graph
blowup when colored with a fractional positive semidefinite forcing set are
examined and used to define a three-color forcing game that directly computes
the fractional positive semidefinite forcing number of a graph. We develop a
fractional parameter based on the standard zero forcing process and it is shown
that this parameter is exactly the skew zero forcing number with a three-color
approach. This approach and an algorithm are used to characterize graphs whose
skew zero forcing number equals zero.Comment: 24 page
Zero forcing number: Results for computation and comparison with other graph parameters
The zero forcing number of a graph is the minimum size of a zero forcing set. This parameter is useful in the minimum rank/maximum nullity problem, as it gives an upper bound to the maximum nullity. Results for determining graphs with extreme zero forcing numbers, for determining the zero forcing number of a graph with a cut-vertex, and for determining the zero forcing number of unicyclic graphs are presented. The zero forcing number is compared to the path cover number and the maximum nullity with equality of zero forcing number and path cover number shown for all cacti and equality of zero forcing number and maximum nullity shown for a subset of cacti
Forts, (fractional) zero forcing, and Cartesian products of graphs
The (disjoint) fort number and fractional zero forcing number are introduced
and related to existing parameters including the (standard) zero forcing
number. The fort hypergraph is introduced and hypergraph results on
transversals and matchings are applied to the zero forcing number and fort
number. These results are used to establish a Vizing-like lower bound for the
zero forcing number of a Cartesian product of graphs for certain families of
graphs, and a family of graphs achieving this lower bound is exhibited
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