Fort Cover Model in Graph Theory

• Fort Cover Model is a mathematical framework that uses integer programming to compute minimum zero forcing sets by covering forts defined through specific adjacency properties.
• The model reformulates the zero forcing problem as a covering problem, allowing both exact and fractional relaxations that yield efficient lower bounds on key graph parameters.
• Practical implementations incorporate constraint generation techniques to compute additional invariants such as propagation time and maximum nullity, supporting theoretical conjectures in graph theory.

The Fort Cover Model provides an integer programming framework for computing minimum zero forcing sets and related graph-theoretic parameters through the concept of "forts"—special vertex subsets defined by their adjacency properties. Originating in zero forcing theory, the model reformulates the combinatorial zero forcing number problem as a covering problem over all forts, enabling both exact and fractional relaxations and supporting the computation of further graph invariants such as propagation time and maximum nullity.

1. Zero Forcing Sets and the Definition of Forts

Zero forcing is an iterative coloring process on a graph $G=(V,E)$: starting with a subset of vertices colored blue, the color-change rule allows a blue vertex with exactly one white (unfilled) neighbor to force it blue. The smallest such initial blue set capable of coloring the entire graph is the zero forcing set; its cardinality is the zero forcing number $Z(G)$. Complementarily, a fort is defined as a nonempty subset $F \subseteq V$ such that no vertex outside $F$ has exactly one neighbor in $F$. The central theorem underlying the Fort Cover Model asserts that an initial set $S$ is a zero forcing set if and only if it intersects every fort in $G$, i.e., $S$ covers all forts.

2. Integer Programming Formulation

The Fort Cover Model translates the combinatorial criterion into an integer program. Each vertex $v \in V$ has a binary variable $s_v$ indicating its inclusion in the candidate zero forcing set:

$$\begin{align*} \text{minimize} & \quad \sum_{v \in V} s_v \ \text{subject to} & \quad \sum_{v \in F} s_v \geq 1 \qquad \forall \text{ fort } F \subseteq V \ & \quad s_v \in \{0,1\} \qquad \forall v \in V \end{align*}$$

Any feasible solution to the above system corresponds to a valid zero forcing set, and the optimal value gives $Z(G)$ (Cameron et al., 10 Aug 2025).

3. Extensions: Fractional Zero Forcing Number, Fort Number, and Additional Parameters

Relaxing $s_v$ to $[0,1]$ defines the fractional zero forcing number $Z^*(G)$, which is conjectured to provide a lower bound on the graph's maximum nullity and yields tighter relaxations compared to related IP models.

The fort number $\mathsf{ft}(G)$ is defined as the maximum number of pairwise disjoint forts. The paper establishes the bounds:

$\mathsf{ft}(G) \leq Z^*(G) \leq Z(G)$

The Fort Cover Model is also leveraged to compute additional parameters: realized propagation time intervals, all minimal forts, the fort number, and throttling numbers. Although direct time-step or infection dynamics are not encoded in the Fort Cover Model itself, its constraint structure allows for the derivation of strong inequalities for use in dynamic models—thereby supporting the computation of propagation times and throttling numbers when integrated into broader frameworks.

4. Constraint Generation Techniques and Practical Computation

Since the number of minimal forts in a graph can be exponential, constraint generation (cutting plane methods) is essential. The algorithm proceeds by starting with a small set of explicit fort constraints, solving the relaxed system, and using a dedicated "Minimum Fort Model" to identify uncovered forts in the current solution:

• If a violated fort $F$ is found (i.e., $\sum_{v \in F} s_v < 1$), add its cover constraint.
• Repeat until all forts are covered.

The separation problem—identifying uncovered or low-weight forts—is itself formulated via integer programming and solved recursively or via specialized algorithms for trees and other graph classes.

5. Numerical Experiments and Empirical Findings

Computational comparisons in the paper show that on small and medium graphs (order $4$–$9$), the Fort Cover Model is competitive with other IP approaches such as the Infection Model and Time Step Model, especially in its LP relaxation form. The LP relaxation of the Fort Cover Model is notably efficient and produces lower bounds for $Z(G)$. Specialized algorithms for fort enumeration yield new insights into the number of minimal forts in various graph classes, informing conjectures about their growth rates (e.g., for trees the ratio of minimal fort counts approaches the plastic ratio).

Empirical results support several open conjectures, specifically:

• For hypercubes $Q_d$, the full realized propagation time interval is confirmed for $d=2,3,4,5$.
• For trees, the plastic ratio emerges in the count of minimal forts for extremal cases.
• For certain known graphs, calculated bounds suggest $Z^*(G)$ and $\mathsf{ft}(G)$ offer robust lower bounds on maximum nullity, complementing algebraic methods.

6. Summary and Connections to Broader Graph Theory

The Fort Cover Model reinterprets the zero forcing set problem through the lens of set covering in combinatorial optimization, where the system of fort constraints captures all minimum zero forcing requirements. Its integer programming and fractional relaxations anchor it within computational graph theory, with application to maximum nullity estimation and related linear algebraic invariants. The model's flexibility permits integration with other parameter estimation procedures, while empirical findings continually inform its effectiveness and theoretical properties.

7. Open Problems, Future Directions, and Impact

Current research directions focus on improving fort enumeration methods, deepening the connection between fractional zero forcing numbers and matrix nullity, and exploring the dynamics of propagation time and throttling across graph families. The Fort Cover Model presents a rigorous structure for future computational and theoretical advancements, with several conjectures—such as the tightness of bounds on nullity and the behavior of minimal fort counts—under active investigation through IP-based and combinatorial approaches (Cameron et al., 10 Aug 2025).

Table 1 summarizes the key constraints and parameters associated with this model:

Parameter	Definition	Bound Relationships
Zero Forcing Set	Initial vertices covering all forts; minimum under set cover IP	$Z(G) \geq Z^*(G) \geq \mathsf{ft}(G)$
Fort	Subset $F$ not exposed to unique neighbor outside $F$	Used in covering constraints
Fractional ZF	LP relaxation of fort covering constraints	Lower bound for maximum nullity conjectured
Fort Number	Maximum cardinality of disjoint fort family	Lower bound for $Z^*(G)$

The Fort Cover Model thus constitutes a central framework in static zero forcing optimization, furnishing both new theoretical insights and efficient computational tools in graph theory.