The zero forcing polynomial of a graph

Abstract

Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored vertices which causes the entire graph to eventually become colored. In this paper, we study the counting problem associated with zero forcing. We introduce the zero forcing polynomial of a graph G of order n as the polynomialis the number of zero forcing sets of G of size i. We characterize the extremal coefficients of Z(G; x), derive closed form expressions for the zero forcing polynomials of several families of graphs, and explore various structural properties of Z(G; x), including multiplicativity, unimodality, and uniqueness.

Isomorphisms and properties of TAR reconfiguration graphs for zero forcing and other $X$-set parameters

“…Let G be a graph of order n and let z(G; k) denote the number of zero forcing sets of cardinality k. Boyer et al defined the zero forcing polynomial of G to be Z(G; x) = n k=Z(G) z(G; k)x k in [5]. The next result is immediate from Corollary 2.11 (or Corollary 2.13).…”

“…We show that for graphs G and G ′ with no isolated vertices, X TAR (G) ∼ = X TAR (G ′ ) implies there is a relabeling of the vertices of G ′ such that G and G ′ have exactly the same X-sets (see Theorem 2.12). When applied to zero forcing, Z TAR (G) ∼ = Z TAR (G ′ ) implies G and G ′ necessarily have the same the zero forcing polynomials (as defined in [5], see Section 2.3). Showing that an isomorphism of zero forcing TAR graphs implies that their base graphs have the same the zero forcing polynomial triggered this study of isomorphisms.…”

Zero-forcing in random regular graphs

“…Then we let vector-valued function y be the solution to a differential equation similar to (5). Analogously to the analysis of Algorithm 1 for d = 3 there are two phases.…”

“…Many facets and generalizations of zero forcing have been studied. Some of note are the propagation time, which is defined to be the minimum amount of time for a minimum zero forcing set to force the entire graph, where at each time step all possible forces happen simultaneously [13], positive semi-definite zero forcing, a variant which bounds the minimum rank problem when the minimum is taken only over positive semi-definite matrices with graph G [8], and the zero forcing polynomial, which algebraically encodes information about the number of zero forcing sets of a given size in a graph [5].…”

Minimal Zero Forcing Sets

“…Maximum minimal sets and minimum maximal sets have been studied in the context of many other graph parameters, including independent sets (see [17,19]), dominating sets (see [5,6]), matchings (see [15,16]), and vertex covers (see [8,25]). Studying minimal zero forcing sets can lead to a better understanding of the zero forcing process, e.g., in the context of zero forcing polynomials [9] and zero forcing reconfiguration graphs [18].…”