Fractional Zero Forcing Number in Graphs

• Fractional Zero Forcing Number is a real-valued graph parameter that relaxes the traditional zero forcing process through linear programming formulations.
• It bridges the gap between classical zero forcing and fort-based bounds by establishing a hierarchy between the fort number and the standard zero forcing number.
• The parameter provides actionable insights for computing bounds and developing efficient LP/IP models for combinatorial optimization in graph theory.

The fractional zero forcing number is a real-valued graph parameter arising from the relaxation of the zero forcing process—a discrete propagation game with applications to linear algebraic graph invariants and combinatorial optimization. It quantifies the minimal “fractional” size of a set needed to force all vertices of a graph gray under the relaxed analogue of the zero forcing rule, fundamentally connecting fort packing and transversal concepts within hypergraph theory. The fractional zero forcing number unifies minimum zero forcing sets, fort-based lower bounds, and provides a hierarchy between fort number and standard zero forcing number.

1. Formal Definition and Underlying Structures

Given a simple undirected graph $G=(V,E)$, a set $C\subseteq V$ is a zero forcing set if, iteratively applying the color-change rule (any gray vertex with exactly one white neighbor forces that neighbor to gray), every vertex is eventually forced gray. The zero forcing number $Z(G)$ is the minimal size of such a set.

A fort is a nonempty $F\subseteq V$ such that no vertex outside $F$ has exactly one neighbor in $F$; denote by $\mathcal{F}_G$ the collection of all minimal forts. A fundamental characterization holds: $S\subseteq V$ is a zero forcing set if and only if $S$ intersects every fort ($S\cap F\neq \emptyset\ \forall F\in\mathcal{F}_G$).

The fractional zero forcing number, denoted $C\subseteq V$0 or $C\subseteq V$1, is the optimum value of the linear program: $C\subseteq V$2 This is the fractional transversal number $C\subseteq V$3 of the fort-hypergraph $C\subseteq V$4, where each minimal fort forms a hyperedge.

Dually, the fractional fort-number $C\subseteq V$5 is the fractional matching number of $C\subseteq V$6: $C\subseteq V$7 By duality, $C\subseteq V$8 (Cameron et al., 2023).

2. Relationships Among Z(G), fort number, and Z⁎(G)

The parameters satisfy the sandwich inequality: $C\subseteq V$9 where $Z(G)$0 is the size of the largest collection of pairwise-disjoint forts (packing number for the fort-hypergraph). The standard zero forcing number is the integral transversal number $Z(G)$1, and the fort number is the integral matching number $Z(G)$2 (Cameron et al., 2023).

If all minimal forts are pairwise-disjoint, $Z(G)$3 holds.

3. Linear and Integer Programming Models

Three principal integer programming (IP) and linear programming (LP) models are standard for $Z(G)$4:

Model	Objective	Constraints
Fort Cover (IP)	$Z(G)$5	$Z(G)$6, $Z(G)$7
Fort Cover (LP)	$Z(G)$8	$Z(G)$9, $F\subseteq V$0
Dual Fort Packing	$F\subseteq V$1	$F\subseteq V$2, $F\subseteq V$3

The LP solution gives $F\subseteq V$4, the IP solution gives $F\subseteq V$5, and the dual LP gives $F\subseteq V$6 (Cameron et al., 10 Aug 2025).

4. Hypergraph Bounds and Cartesian Product Behavior

Hypergraph transversal and matching theory yield several bounds:

• If every minimal fort has size $F\subseteq V$7 and the fort-hypergraph contains $F\subseteq V$8 vertices,

$F\subseteq V$9

• If each vertex appears in at most $F$0 minimal forts and there are $F$1 minimal forts,

$F$2

• For connected $F$3,

$F$4

For Cartesian products, if $F$5 and $F$6 are graphs each with at least one edge,

$F$7

If $F$8, then

$F$9

These results provide both lower and upper bounds for fractional zero forcing under graph operations (Cameron et al., 2023).

5. Fractional Zero Forcing, Forcing Games, and Three-Color Approaches

Alternative characterizations of the fractional zero forcing number utilize multi-color forcing games and graph blowups. An $F$0-fold blowup replaces each vertex of $F$1 with $F$2 independent vertices, modeling the propagation process over a larger structure.

The $F$3-fold zero forcing number $F$4 is defined via simultaneous forces on the blowup $F$5; the fractional forcing number is then $F$6 (Hogben et al., 2015).

A three-color game (skew zero forcing) uses colors: dark blue (target), light blue, and white, with the rule: if a vertex $F$7 (either dark or light blue) has exactly one non-dark-blue neighbor, $F$8 can force it to dark blue. The fractional parameter realized by this game is equal to $F$9.

For every graph $\mathcal{F}_G$0, the following equality holds: $\mathcal{F}_G$1 Thus, the fractional zero forcing number coincides with the skew zero forcing number derived from these game-theoretic formulations (Hogben et al., 2015).

6. Example Calculations and Extremal Values

Specific graph families illustrate the diversity of $\mathcal{F}_G$2:

Graph	$\mathcal{F}_G$3	$\mathcal{F}_G$4	$\mathcal{F}_G$5
Path $\mathcal{F}_G$6	$\mathcal{F}_G$7	$\mathcal{F}_G$8	$\mathcal{F}_G$9
Even cycle $S\subseteq V$0	$S\subseteq V$1	$S\subseteq V$2	$S\subseteq V$3
Odd cycle $S\subseteq V$4	$S\subseteq V$5	$S\subseteq V$6	$S\subseteq V$7
Complete $S\subseteq V$8	$S\subseteq V$9	$S$0	$S$1
$S$2, $S$3	$S$4	$S$5	$S$6
Corona $S$7	$S$8	$S$9	$S\cap F\neq \emptyset\ \forall F\in\mathcal{F}_G$0

Extremal properties include:

• $S\cap F\neq \emptyset\ \forall F\in\mathcal{F}_G$1 if and only if $S\cap F\neq \emptyset\ \forall F\in\mathcal{F}_G$2 is a path.
• $S\cap F\neq \emptyset\ \forall F\in\mathcal{F}_G$3 if every vertex lies in some 2-element fort (Cameron et al., 2023).

7. Open Problems and Research Directions

Principal open questions for the fractional zero forcing number include:

• The existence of a purely combinatorial forcing game whose integer relaxation yields $S\cap F\neq \emptyset\ \forall F\in\mathcal{F}_G$4 remains unknown (Cameron et al., 2023).
• The relationship between maximum nullity $S\cap F\neq \emptyset\ \forall F\in\mathcal{F}_G$5, fort number $S\cap F\neq \emptyset\ \forall F\in\mathcal{F}_G$6, and $S\cap F\neq \emptyset\ \forall F\in\mathcal{F}_G$7: whether $S\cap F\neq \emptyset\ \forall F\in\mathcal{F}_G$8 or $S\cap F\neq \emptyset\ \forall F\in\mathcal{F}_G$9 for all $C\subseteq V$00 is unresolved.
• Patterns of equality in certain Cartesian product families, and possible characterizations of graphs achieving the Vizing-type lower bound, present ongoing areas for combinatorial investigation (Cameron et al., 2023).
• Application of fractional zero forcing to efficient computation in medium-sized graphs using IP/LP models, as well as numerical exploration of conjectured parameter bounds (Cameron et al., 10 Aug 2025).

Ongoing research combines combinatorial, linear-optimization, and algebraic perspectives, with connections to hypergraph theory, spectral graph theory, and coloring games.