Zero Forcing Number in Graph Theory

• Zero forcing number is a graph invariant defined as the minimum size of a vertex set that forces the entire graph to black using a unique neighbor rule.
• It serves as an upper bound on maximum nullity, linking combinatorial graph properties with matrix theory and minimum rank problems.
• Variants like positive semidefinite zero forcing and integer programming models extend its applications to network control, quantum systems, and optimization.

The zero forcing number, denoted $Z(G)$ for a graph $G$, is a combinatorial graph invariant defined as the minimum cardinality of a subset $Z \subseteq V(G)$ such that, starting with all vertices in $Z$ colored black and the remainder white, iterative applications of the color change rule ("if a black vertex has exactly one white neighbor, force that neighbor to black") result in the entire vertex set being black. Zero forcing was introduced to provide a combinatorial bound on the maximum nullity (or, equivalently, on the minimum rank) of real symmetric matrices associated with the graph's adjacency pattern, but its scope has expanded to include connections to positive semidefinite rank, quantum controllability, network propagation, and algebraic graph theory (Barioli et al., 2010).

1. Formal Definition and Core Properties

The zero forcing process on a finite, simple, undirected graph $G = (V,E)$ proceeds as follows:

• An initial subset $Z \subseteq V$ is chosen and colored black, all other vertices are white.
• At each step, if a black vertex $u$ has exactly one white neighbor $w$, then the "color change rule" forces $w$ to turn black (denoted $u \to w$).
• The iterative procedure continues until no further vertices can be forced.

A subset $G$0 such that this process results in all vertices of $G$1 being black is called a zero forcing set. The zero forcing number is defined by

$G$2

A fundamental property is that for any $G$3-vertex graph $G$4, $G$5 (the minimum degree) (Eroh et al., 2014). In connected graphs of order at least two, no vertex belongs to every minimum zero forcing set; that is,

$G$6

(Barioli et al., 2010).

2. Connections with Matrix Theory and Rank Problems

A key impact of the zero forcing number is as an upper bound on the maximum nullity $G$7 of the family of real symmetric matrices with off-diagonal zero–nonzero pattern described by $G$8: $G$9 This connection comes from the "color-change" rule mirroring restrictions imposed on vectors in the kernel of matrices associated with $Z \subseteq V(G)$0 (Barioli et al., 2010). For positive semidefinite matrices, an adapted forcing process leads to the positive semidefinite zero forcing number $Z \subseteq V(G)$1, which also satisfies $Z \subseteq V(G)$2. For real symmetric and complex Hermitian matrix families, differences between $Z \subseteq V(G)$3 and $Z \subseteq V(G)$4 highlight the algebraic subtleties arising from field choice, as in the example of the $Z \subseteq V(G)$5-wheel with 4 hubs where $Z \subseteq V(G)$6 but $Z \subseteq V(G)$7 (Barioli et al., 2010).

3. Positive Semidefinite Zero Forcing and Ordered Set Number

The positive semidefinite zero forcing number $Z \subseteq V(G)$8 is based on a modified rule: after blacking $Z \subseteq V(G)$9, write $Z$0 with components $Z$1; if $Z$2, and in $Z$3 vertex $Z$4 has a unique white neighbor $Z$5 in $Z$6, then $Z$7 forces $Z$8. The number $Z$9 is then the minimum size of such a set to force the entire graph black.

It is characterized in terms of the ordered set number $G = (V,E)$0 (a parameter derived from certain vertex orderings constrained by adjacency). The key result is

$G = (V,E)$1

where $G = (V,E)$2 is the order of $G = (V,E)$3 (Barioli et al., 2010). This relation allows $G = (V,E)$4 to be determined via purely combinatorial means, and is especially effective in families such as trees, generalized book graphs, and various Cartesian products.

4. Structural Results, Extremal Examples, and Variants

Examples provided in the literature illustrate the sensitivity of $G = (V,E)$5 and $G = (V,E)$6 to graph structure:

• For the 12-vertex "pinwheel" outerplanar 2-tree $G = (V,E)$7, $G = (V,E)$8 but $G = (V,E)$9; both maximum nullity and path cover number are 3, showing tight relations among invariants.
• In $Z \subseteq V$0-wheel graphs $Z \subseteq V$1, real vs. complex field effects result in different possible minimum ranks for positive semidefinite matrices (Barioli et al., 2010).

No vertex lies in every minimum zero forcing set of a connected graph with at least two vertices, a property established by observing that minimum zero forcing sets can be "reversed" via forcing paths and chains, demonstrating the high degree of non-uniqueness and flexibility in zero forcing processes.

5. Bounds, Characterizations, and Graph Families

Zero forcing numbers admit various general bounds and have been characterized for specific graph classes:

• For any connected graph of order $Z \subseteq V$2, $Z \subseteq V$3 if its complement is also connected; more generally,

$Z \subseteq V$4

with both bounds being sharp (Eroh et al., 2014).

• For block-cycle graphs, $Z \subseteq V$5 equals the path cover number $Z \subseteq V$6; for outerplanar graphs, $Z \subseteq V$7 equals the tree cover number $Z \subseteq V$8 (Taklimi et al., 2013, Taklimi, 2013).
• In random graphs and pseudorandom graphs, spectral methods and expansion properties yield asymptotic estimates: $Z \subseteq V$9 for Erdős–Rényi graphs (Kalinowski et al., 2017), and

$u$0

in $u$1-regular graphs (where $u$2 is the smallest eigenvalue) (Kalinowski et al., 2017).

6. Algorithmic Aspects and Integer Programming

Computing $u$3 is NP-hard for general graphs (Trefois et al., 2014). Several exact integer programming models have been proposed, including:

• The Infection Model and Time Step Model, which encode forcing sequences and propagation timing, allowing determination of not only $u$4 but also minimum/maximum propagation times and the throttling parameter (set size plus propagation time).
• The Fort Cover Model, based on the concept of forts (non-empty induced subsets where no outside vertex has exactly one neighbor in the set), leverages set cover structure to model zero forcing and its fractional relaxations. The fractional zero forcing number $u$5 (LP relaxation of the Fort Cover Model) and the fort number $u$6 (the maximal number of pairwise disjoint forts) satisfy $u$7, potentially providing new lower bounds on maximum nullity (Cameron et al., 10 Aug 2025).

Numerical experiments confirm the tractability of these models for small/medium graphs, and support conjectures regarding realized propagation times, ratios of minimal fort counts across tree orders, and lower bounds relating $u$8, $u$9, and $w$0.

7. Applications, Extensions, and Open Directions

Zero forcing numbers have interpreted significance in numerous contexts:

• Minimum Rank Problems: $w$1 provides a powerful combinatorial tool for bounding matrix nullities and characterizing forbidden subgraphs via tight inequalities.
• Network Control and Quantum Controllability: The underlying dynamic of the color-change rule parallels the spread of control or infection in physical and cyber systems; the size and structure of minimal zero forcing sets dictate the resource requirements for full-state control (Barioli et al., 2010).
• Graph Coverings: Their equivalence to classic covering parameters (path cover, tree cover) in certain families connects zero forcing to established domains in combinatorial optimization (Taklimi et al., 2013, Taklimi, 2013).
• Graph Inertia: Variants such as the $w$2-forcing game yield systematic methods for bounding inertia sets of graphs, providing insights on possible eigenvalue multiplicity combinations for matrix families indexed by a graph (Butler et al., 2012).
• Hierarchy/Product Structures: Extensions to generalized hierarchical products offer new bounds on $w$3 in complex graph constructions such as Cartesian and hierarchical products, relevant in modular or hierarchical network architectures (LeClair et al., 2024).

Further research directions include refinement of combinatorial bounds, classification of extremal graphs with prescribed forcing numbers, exploration of field-dependent phenomena in matrix rank via zero forcing parameters, and algorithmic improvements for computing or approximating zero forcing and related invariants. The interplay between algebraic, combinatorial, and probabilistic techniques continues to drive advances in the understanding of zero forcing and its extensions.