Zero Blocking Number in Graph Theory

• Zero Blocking Number is a graph parameter defined through the zero forcing process that quantifies the smallest set whose presence stops complete black propagation.
• It is closely linked with the failed zero forcing number, with exact formulas known for paths, cycles, hypercubes, grids, and several set-intersection graphs.
• Recent research provides structural characterizations, NP-completeness results, and efficient dynamic programming algorithms for trees, alongside applications in linear algebra and game theory.

The zero blocking number is a graph parameter attached to the zero forcing process. In that process, vertices are initially colored black or white, and whenever a black vertex has exactly one white neighbor, that white neighbor is forced to become black. A zero blocking set is an initial white set whose presence prevents the process from turning every vertex black, and the zero blocking number $B(G)$ is the minimum size of such a set. Equivalently, if $F(G)$ denotes the failed zero forcing number, then $F(G)+B(G)=|V(G)|$. Recent work gives exact values of $B(G)$ for several major graph families, including generalized Kneser graphs, generalized Johnson graphs, grid graphs, hypercubes, unions, joins, and trees, while also establishing NP-Completeness for the general computation problem (Lin et al., 25 Aug 2025, Lin et al., 4 Aug 2025, Lin et al., 13 Aug 2025).

1. Zero forcing, failed forcing, and blocking

A zero forcing process starts from a prescribed black set and repeatedly applies the color-change rule: if a black vertex has exactly one white neighbor, that white neighbor becomes black. A zero forcing set is an initial black set that eventually forces every vertex to black. A failed zero forcing set is an initial black set that does not achieve this, and a zero blocking set is its complement. One formulation states that a zero blocking set is the complement of some failed zero forcing set; another states that it is a set of white vertices that cannot all be forced to black. The associated minimum is

$$B(G)=\min\{\,|W| \mid W \subseteq V(G)\text{ is a zero blocking set of }G\,\}.$$

The relation with failed zero forcing is

$B(G)=|V(G)|-F(G).$

A useful terminal notion is the final zero blocking set: a nonempty white set $W\subset V$ such that no black vertex is adjacent to exactly one white vertex. In that stalled configuration, the forcing process cannot continue. This formulation is central in both structural and algorithmic treatments of $B(G)$ (Lin et al., 25 Aug 2025).

2. Structural criteria and exact values for standard graph families

Several elementary criteria sharply constrain $B(G)$. If a graph $G$ has an isolated vertex, then $F(G)$0; otherwise $F(G)$1. If $F(G)$2 has components $F(G)$3, then

$F(G)$4

A particularly clean characterization occurs at value $F(G)$5: for a graph without isolated vertices, $F(G)$6 if and only if $F(G)$7 has a pair of twins, where distinct vertices $F(G)$8 are twins if $F(G)$9 or $F(G)+B(G)=|V(G)|$0 (Lin et al., 25 Aug 2025).

Exact formulas are known for several basic families. For a path $F(G)+B(G)=|V(G)|$1 and cycle $F(G)+B(G)=|V(G)|$2,

$F(G)+B(G)=|V(G)|$3

For the $F(G)+B(G)=|V(G)|$4-dimensional hypercube $F(G)+B(G)=|V(G)|$5 with $F(G)+B(G)=|V(G)|$6,

$F(G)+B(G)=|V(G)|$7

Moreover, the minimum zero blocking sets in $F(G)+B(G)=|V(G)|$8 are exactly the neighborhoods $F(G)+B(G)=|V(G)|$9 of single vertices $B(G)$0; when $B(G)$1, there are also minimum sets of the form $B(G)$2 for vertices $B(G)$3 at distance $B(G)$4 (Lin et al., 25 Aug 2025).

Graph operations admit exact formulas as well. For unions,

$B(G)$5

For joins, the description uses the second zero blocking number $B(G)$6, defined as the minimum size of a zero blocking set of size at least two:

$B(G)$7

If $B(G)$8 and $B(G)$9, then

$$B(G)=\min\{\,|W| \mid W \subseteq V(G)\text{ is a zero blocking set of }G\,\}.$$0

where

$$B(G)=\min\{\,|W| \mid W \subseteq V(G)\text{ is a zero blocking set of }G\,\}.$$1

with $$B(G)=\min\{\,|W| \mid W \subseteq V(G)\text{ is a zero blocking set of }G\,\}.$$2 the domination number. Corollaries include the facts that all cographs of at least two vertices have twins and hence satisfy $$B(G)=\min\{\,|W| \mid W \subseteq V(G)\text{ is a zero blocking set of }G\,\}.$$3 unless there is an isolated vertex, and that for wheel graphs $$B(G)=\min\{\,|W| \mid W \subseteq V(G)\text{ is a zero blocking set of }G\,\}.$$4,

$$B(G)=\min\{\,|W| \mid W \subseteq V(G)\text{ is a zero blocking set of }G\,\}.$$5

except that $$B(G)=\min\{\,|W| \mid W \subseteq V(G)\text{ is a zero blocking set of }G\,\}.$$6 (Lin et al., 25 Aug 2025).

3. Complexity and exact computation on trees

The global computation problem for zero blocking number is intractable in general. Computing $$B(G)=\min\{\,|W| \mid W \subseteq V(G)\text{ is a zero blocking set of }G\,\}.$$7 is NP-Complete, and the hardness persists even on chordal and bipartite graphs. An equivalent decision problem is:

Does $$B(G)=\min\{\,|W| \mid W \subseteq V(G)\text{ is a zero blocking set of }G\,\}.$$8 contain a (final) zero blocking set of size at most $$B(G)=\min\{\,|W| \mid W \subseteq V(G)\text{ is a zero blocking set of }G\,\}.$$9?

Against that hardness backdrop, trees admit a linear-time dynamic programming algorithm. The method roots the tree at a chosen node and, for each rooted subtree $B(G)=|V(G)|-F(G).$0, tracks three quantities:

• $B(G)=|V(G)|-F(G).$1: minimum size of a final zero blocking set of $B(G)=|V(G)|-F(G).$2 excluding $B(G)=|V(G)|-F(G).$3;
• $B(G)=|V(G)|-F(G).$4: minimum size of a final zero blocking set of $B(G)=|V(G)|-F(G).$5 including $B(G)=|V(G)|-F(G).$6;
• $B(G)=|V(G)|-F(G).$7: minimum size of a final $B(G)=|V(G)|-F(G).$8-zero blocking set, where $B(G)=|V(G)|-F(G).$9 is black but has at least one white neighbor and the others cannot force.

When composing rooted trees by attaching a child $W\subset V$0 to $W\subset V$1, the recurrences are

$W\subset V$2

$W\subset V$3

$W\subset V$4

The algorithm, denoted zbsTreeD, roots the tree arbitrarily, processes vertices in post-order, updates these states along parent-child links, and returns

$W\subset V$5

The paper also discusses the extension of this DP approach to block graphs (Lin et al., 25 Aug 2025).

4. Generalized Kneser graphs and generalized Johnson graphs

A major recent extension concerns zero blocking numbers for generalized set-intersection graphs. For integers $W\subset V$6, the generalized Kneser graph $W\subset V$7 has as vertices the $W\subset V$8-subsets of $W\subset V$9, with adjacency defined by $B(G)$0. The generalized Johnson graph $B(G)$1 has the same vertex set, with adjacency defined by $B(G)$2. Standard Kneser graphs arise as $B(G)$3, and standard Johnson graphs as $B(G)$4. The recent exact results extend the theorem of Afzali, Ghodrati, and Maimani, which gave $B(G)$5 for $B(G)$6, $B(G)$7, and $B(G)$8 for $B(G)$9, $B(G)$0, except $B(G)$1 (Lin et al., 4 Aug 2025).

For generalized Kneser graphs, if $B(G)$2 and $B(G)$3, then

$B(G)$4

This recovers the ordinary Kneser formula when $B(G)$5, and, for example, yields $B(G)$6 for sufficiently large $B(G)$7. The paper also characterizes several small-parameter cases:

• if $B(G)$8, then $B(G)$9;
• if $G$0, then $G$1;
• if $G$2 and $G$3, then $G$4.

For generalized Kneser graphs there is also a monotonicity statement: if $G$5, then

$G$6

and a universal upper bound

$G$7

For generalized Johnson graphs, the exact value depends on the regime of $G$8. If $G$9 and $F(G)$00, then

$F(G)$01

If $F(G)$02 and $F(G)$03, then

$F(G)$04

Equivalently, for sufficiently large $F(G)$05, the formula may be summarized as

$F(G)$06

The general upper bound is

$F(G)$07

for any $F(G)$08. Boundary cases are explicitly characterized:

• if $F(G)$09, then $F(G)$10;
• if $F(G)$11 and $F(G)$12, or $F(G)$13, then $F(G)$14;
• if $F(G)$15 and $F(G)$16, then $F(G)$17;
• if $F(G)$18, then $F(G)$19.

The proofs extend the techniques used for ordinary Kneser and Johnson graphs by analyzing intersection patterns among $F(G)$20-subsets, constructing zero blocking sets from families of pairwise-disjoint $F(G)$21-subsets, and using double counting and inclusion-exclusion to exclude smaller blocking configurations. For Johnson graphs, symmetry and the relation between $F(G)$22 and its dual $F(G)$23 are used to transfer results between different parameter regimes. The results are stated to be tight for large $F(G)$24, although the given lower bounds on $F(G)$25 are often more than strictly necessary. The paper further states that the exact zero blocking number is now explicitly determined for generalized Kneser and generalized Johnson graphs for all parameters with sufficiently large $F(G)$26, yielding a unified framework for these classical families (Lin et al., 4 Aug 2025).

5. Exact formula for grid graphs

For the grid graph $F(G)$27 with $F(G)$28, the zero blocking number is known exactly. Writing

$F(G)$29

with integers $F(G)$30 satisfying $F(G)$31, the formula is

$F(G)$32

Here

$F(G)$33

The same result is also expressed in alternative forms. If $F(G)$34, then

$F(G)$35

whereas if $F(G)$36, then

$F(G)$37

The proof is described as combinatorial and geometric. It uses a careful characterization of minimum zero blocking sets through their interactions with rows and columns, geometric language involving points, rays, and line segments on the lattice, and case splits based on the arrangement of white vertices. The result confirms that earlier upper bounds are exact and closes an open question. The paper also notes applications of zero blocking to linear algebra, specifically matrix minimum rank problems, and to quantum control theory (Lin et al., 13 Aug 2025).

6. Distinct uses of blocking terminology outside the graph parameter

The phrase blocking number also appears in related but distinct settings. In Blocking-$F(G)$38 Wythoff Nim, the blocking number is the parameter $F(G)$39 governing how many options the previous player may forbid before the next move. The zero blocking number case is $F(G)$40, where no move can be blocked and the game reduces to classical Wythoff Nim. In that case the $F(G)$41-positions are

$F(G)$42

with $F(G)$43 the Golden Ratio. For arbitrary $F(G)$44, a position is $F(G)$45 iff strictly fewer than $F(G)$46 of its options are $F(G)$47-positions. This usage is game-theoretic rather than graph-theoretic, but it employs closely related blocking language (Larsson, 2010).

A second distinct usage occurs for 123-avoiding permutation matrices. There, a blocker is the set of zero positions in a $F(G)$48-forcing matrix, and the zero blocking number is the minimal number of zeros in any such matrix. The cited result is that this number equals $F(G)$49. The paper further introduces flag-shaped blockers $F(G)$50, generalizing $F(G)$51-shaped blockers, with cardinality

$F(G)$52

where $F(G)$53 and $F(G)$54. All flag-shaped blockers are minimum blockers in that setting. This notion of zero blocking number is therefore combinatorial but not derived from zero forcing on graphs (Bennett et al., 2023).