Fort Number in Graph Theory

• Fort number is defined as the maximum number of pairwise disjoint forts, representing static obstructions in the zero forcing process.
• It bridges covering and matching in the fort hypergraph framework and has a dual fractional formulation equal to the fractional zero forcing number.
• Key results include the bound ft(G) ≤ Z(G) ≤ (pw(G)+1)·ft(G) and notable behavior under Cartesian products and specific cases like hypercubes.

In contemporary graph theory, the fort number usually denotes the packing parameter $\ft(G)$ associated with zero forcing: for a finite simple graph $G$, a nonempty set $F\subseteq V(G)$ is a fort if no vertex outside $F$ has exactly one neighbor in $F$, and $\ft(G)$ is the maximum number of pairwise disjoint forts in $G$. This parameter packages the obstruction structure behind zero forcing, because a set is zero forcing if and only if it intersects every fort; consequently, $\ft(G)$ is a lower bound on both the fractional and integral zero forcing numbers (Cameron et al., 2023).

1. Definition and obstruction-theoretic meaning

A fort is a nonempty vertex set $F\subseteq V(G)$ satisfying

$$\forall v\in V(G)\setminus F,\qquad |N_G(v)\cap F|\neq 1.$$

Equivalently, every vertex outside $G$0 has either $G$1 or at least $G$2 neighbors in $G$3. In zero forcing language, this is exactly the configuration that prevents progress: if all vertices in $G$4 are already filled or blue, then no outside vertex has a unique white neighbor in $G$5, so no force can enter $G$6 (Cameron et al., 2024).

The fort number is the maximum size of a family of pairwise disjoint forts: $G$7 It is therefore a packing invariant: it measures how many mutually independent obstructions to forcing can coexist in the same graph. The basic zero-forcing characterization

$G$8

immediately yields

$G$9

since any zero forcing set must hit each fort in any disjoint family (Cameron et al., 2023).

This obstruction viewpoint also explains why forts are central rather than auxiliary. A fort is not a dynamic object arising during a forcing process; it is a static certificate that a prescribed complement cannot force the graph. The fort number aggregates these certificates at the level of disjoint packing.

2. Hypergraph, LP, and fractional formulations

A systematic formulation uses the fort hypergraph $F\subseteq V(G)$0, whose vertices are the vertices of $F\subseteq V(G)$1 lying in at least one minimal fort and whose hyperedges are the minimal forts of $F\subseteq V(G)$2. Within this framework, zero forcing and fort packing become standard covering and matching parameters: $F\subseteq V(G)$3 where $F\subseteq V(G)$4 is the transversal number and $F\subseteq V(G)$5 is the matching number of the hypergraph (Cameron et al., 2023).

The same formalism yields the fractional theory. The fractional zero forcing number is

$F\subseteq V(G)$6

and the dual fractional fort number is

$F\subseteq V(G)$7

By linear programming duality,

$F\subseteq V(G)$8

Accordingly, the standard chain is

$F\subseteq V(G)$9

This formulation has two important consequences. First, it places the fort number inside a classical packing-versus-covering paradigm rather than treating it as an ad hoc graph invariant. Second, it clarifies that $F$0 is the integral packing shadow of the same fort system whose transversal number gives $F$1. In the literature this also connects $F$2 with the failed zero forcing partition number via

$F$3

so the parameter has an equivalent interpretation in failed-forcing theory (Cameron et al., 2023).

3. Pathwidth-controlled approximation and the first general gap bound

A major recent development is the pathwidth-sensitive upper bound relating the fort number to the zero forcing number. Given a nice path decomposition of width $F$4, an approximation algorithm constructs simultaneously a zero forcing set $F$5, a forcing arc set $F$6, and a fort packing $F$7. The algorithm processes the graph along the path decomposition and repeatedly uses a vertex-cut gluing proposition for forcing arc sets; because each bag has size at most $F$8, each iteration adds at most $F$9 vertices to $F$0 while also adding one fort to the packing. From this, the paper proves that the output satisfies $F$1, and with an optimal path decomposition obtains

$F$2

The algorithm runs in $F$3 time starting from a path decomposition, where $F$4 and $F$5 are the order and size of the graph (Cameron et al., 2024).

This is significant because it gives the first general upper bound on the multiplicative gap between the fort number and the zero forcing number. Earlier work had established equality only at the fractional level, but no bound of the form $F$6 had been known. The new result shows that pathwidth controls that gap: $F$7

The proof is phrased through forcing arc sets rather than directly through forcing sequences. The technical reason is compositionality under vertex cuts: if two subgraphs share a cut $F$8 and each has a forcing arc set with all vertices of $F$9 as sources, then the arc sets combine into a forcing arc set for the whole graph. This vertex-cut mechanism is the engine behind the approximation ratio (Cameron et al., 2024).

The result also leaves two explicit open directions. One is whether the bound

$\ft(G)$0

is sharp. The other is whether an analogous statement can be proved with tree decompositions, replacing pathwidth by treewidth.

4. Cartesian products and multiplicative behavior

The fort number interacts particularly well with Cartesian products. If $\ft(G)$1 is a fort of $\ft(G)$2 and $\ft(G)$3 is a fort of $\ft(G)$4, then

$\ft(G)$5

is a fort of $\ft(G)$6. This product construction transfers directly to packings and implies

$\ft(G)$7

It is sharp for the family $\ft(G)$8, where

$\ft(G)$9

was proved (Cameron et al., 2023).

The same machinery feeds back into zero forcing. Using fort hypergraphs and hypergraph product inequalities, one obtains

$G$0

In particular, if $G$1, then

$G$2

A stronger lemma yields

$G$3

whenever both graphs contain an edge, and from this comes the Vizing-like lower bound

$G$4

for several graph families, together with the conjecture that the same inequality holds for all graphs with edges (Cameron et al., 2023).

These product results reinforce the interpretation of the fort number as a structural packing invariant. Unlike many lower bounds that are local or degree-based, $G$5 has a clean tensor-like behavior under $G$6, which is precisely why it is useful in zero-forcing product problems.

5. Hypercubes as a model case

The hypercube $G$7 provides the most detailed current case study. In this setting, minimum forts are completely characterized. For $G$8 with $G$9, a set $\ft(G)$0 is a minimum fort if and only if

$\ft(G)$1

for some vertex $\ft(G)$2; hence every minimum fort has size $\ft(G)$3. The exceptional case $\ft(G)$4 has two automorphism classes of minimum forts: the neighborhood forts $\ft(G)$5 and one additional non-neighborhood class represented by a specific $\ft(G)$6-vertex fort. Even there, the minimum fort size remains $\ft(G)$7 (Brimkov et al., 14 Jul 2025).

From this description one gets the exact fractional zero forcing number: $\ft(G)$8 For the fort number itself, the bounds are

$\ft(G)$9

The upper bound comes from $F\subseteq V(G)$0, while the lower bound is obtained from the open packing number because open neighborhoods are forts in $F\subseteq V(G)$1 (Brimkov et al., 14 Jul 2025).

A particularly rigid situation occurs when $F\subseteq V(G)$2. In that case the lower and upper bounds coincide, and

$F\subseteq V(G)$3

Moreover, the paper combines this with known hypercube domination results to obtain

$F\subseteq V(G)$4

The hypercube results are notable for two reasons. First, they show that forts can be much more rigid than zero forcing sets: the paper explicitly contrasts the complete classification of minimum forts with the existence of non-automorphic minimum zero forcing sets having distinct propagation times. Second, they provide one of the clearest examples where the fort number can be bounded sharply and, in the power-of-two case, computed exactly (Brimkov et al., 14 Jul 2025).

6. Minimal forts, computation, and adjacent counting problems

The fort number counts disjoint forts; it does not count all forts or all minimal forts. This distinction matters because minimal forts are the irredundant obstructions used in fort-cover formulations, whereas $F\subseteq V(G)$5 is a packing number. A graph may have exponentially many minimal forts, so the combinatorics of minimal-fort enumeration and the value of the fort number are related but distinct problems (Cameron et al., 10 Aug 2025).

On the optimization side, a dedicated integer program $F\subseteq V(G)$6 computes $F\subseteq V(G)$7 directly without enumerating all minimal forts first. The model uses binary variables $F\subseteq V(G)$8 indicating whether vertex $F\subseteq V(G)$9 belongs to the $$\forall v\in V(G)\setminus F,\qquad |N_G(v)\cap F|\neq 1.$$0-th candidate fort and $$\forall v\in V(G)\setminus F,\qquad |N_G(v)\cap F|\neq 1.$$1 indicating whether that candidate set is nonempty, with objective

$$\forall v\in V(G)\setminus F,\qquad |N_G(v)\cap F|\neq 1.$$2

Its constraints enforce nonemptiness, the fort condition, and pairwise disjointness, and the paper proves that if $$\forall v\in V(G)\setminus F,\qquad |N_G(v)\cap F|\neq 1.$$3 is optimal, then

$$\forall v\in V(G)\setminus F,\qquad |N_G(v)\cap F|\neq 1.$$4

The same paper records the standard hierarchy

$$\forall v\in V(G)\setminus F,\qquad |N_G(v)\cap F|\neq 1.$$5

and discusses the conjectured strengthening

$$\forall v\in V(G)\setminus F,\qquad |N_G(v)\cap F|\neq 1.$$6

in terms of maximum nullity (Cameron et al., 10 Aug 2025).

Adjacent work on minimal forts has revealed a rich counting theory. For graphs of order at least six, the number of minimal forts is strictly less than Sperner’s bound, although minimal forts form a clutter. Exact formulas are known for several graph families: in particular, the number of minimal forts of a path graph follows the Padovan sequence, the number for a cycle follows the Perrin sequence, and windmill graphs $$\forall v\in V(G)\setminus F,\qquad |N_G(v)\cap F|\neq 1.$$7 exhibit exponential growth with asymptotic base $$\forall v\in V(G)\setminus F,\qquad |N_G(v)\cap F|\neq 1.$$8 (Becker et al., 2024). For trees, every tree of order $$\forall v\in V(G)\setminus F,\qquad |N_G(v)\cap F|\neq 1.$$9 has at least $G$00 minimal forts, with equality characterized by graphs of the form $G$01; more recently, the conjectured upper bound

$G$02

for the maximum number of minimal forts on an $G$03-vertex tree was proved (Cameron et al., 14 Dec 2025, Dat et al., 8 May 2026).

These results sharpen a common distinction. The number of minimal forts is an enumeration problem about inclusion-minimal obstructions. The fort number is a packing problem about pairwise disjoint obstructions. The two interact, but they are not interchangeable invariants.

7. Terminological distinctions and scope

In the zero-forcing literature, fort number refers to $G$04, the maximum size of a pairwise disjoint fort packing. However, the phrase Fort numbers has another established meaning in a different combinatorial context: in work on domination in Johnson graphs $G$05, it refers to the Fort–Hedlund covering numbers

$G$06

the minimum number of triangles needed to cover all edges of the complete graph $G$07. Those numbers enter the closed form

$G$08

used to determine $G$09, but they are distinct from the zero-forcing fort number $G$10 (Lee et al., 9 Jun 2026).

This terminological divergence is historically understandable but mathematically substantial. In one setting, the parameter is a graph-packing invariant tied to zero forcing and fort hypergraphs. In the other, it is a triangle-covering number of complete graphs used as an ingredient in domination formulas. For work in zero forcing, spectral graph theory, or propagation processes, the default meaning is the former.

As the subject stands, the fort number occupies a precise niche: it is the integral packing counterpart to the fort-cover description of zero forcing, it admits a fractional LP relaxation identical to fractional zero forcing, it behaves well under Cartesian products, and its gap from $G$11 is now known to be controlled by pathwidth through

$G$12

The current frontier concerns sharpness, treewidth analogues, and broader structural descriptions of graphs where the fort number, fractional zero forcing number, and zero forcing number coincide.