Failed Zero Forcing Number in Graph Theory

• Failed zero forcing number is a graph invariant that measures the largest initial vertex set that fails to color the entire graph under the zero forcing rule.
• It is closely linked with stalled sets and forts, offering a dual perspective to the traditional zero forcing number and aiding in obstruction analysis.
• Extremal bounds, computational complexity, and variants like skew and positive semidefinite failed forcing highlight its rich theoretical and practical implications.

Searching arXiv for recent and foundational papers on failed zero forcing number. Failed zero forcing number is a graph invariant arising from the zero forcing process and measures the opposite extremum from the zero forcing number: instead of asking for the smallest initial set that succeeds in forcing the whole graph, it asks for the largest initial set that still fails. For a graph $G$, a set $S\subseteq V(G)$ is a failed zero forcing set if repeated application of the standard color-change rule does not force all vertices, and the failed zero forcing number is $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$ (Abara et al., 2022). In the equivalent “stalled” formulation used in several papers, one studies proper subsets from which no force is possible; in that language, the largest proper stalled subset defines the same extremal quantity (Shitov, 2016). Subsequent work has developed its complexity theory, extremal bounds, structural dualities with forts, exact values on graph families, variants such as skew and positive semidefinite failed forcing, and behavior on iterated and product graph models (Shitov, 2016, Ufferman et al., 2022, Jacob, 31 Jan 2025, Brice et al., 16 Jul 2025).

1. Definition and basic framework

Let $G=(V,E)$ be a finite simple graph whose vertices are partitioned into filled and empty vertices. The zero forcing rule is: an empty vertex $v$ is forced by a filled vertex $u$ if $v$ is the unique empty neighbor of $u$; equivalently, $u$ is filled and all but one of its neighbors are already filled (Shitov, 2016). In the blue–white formulation, if a blue vertex has exactly one white neighbor, then that white neighbor is forced blue (Abara et al., 2022).

A set $S\subseteq V(G)$ is a zero forcing set if repeated application of the rule eventually fills all vertices, and the zero forcing number $S\subseteq V(G)$0 is the minimum size of such a set (Abara et al., 2022). By contrast, $S\subseteq V(G)$1 is a failed zero forcing set if it does not force all vertices, and

$S\subseteq V(G)$2

This parameter is the failed zero forcing number (Abara et al., 2022).

Several equivalent or closely related notions organize the theory. A set is stalled if no force is possible from it; every stalled set is failed, and every maximum failed set is stalled (Jacob, 31 Jan 2025). In the formulation of failed forcing used for complexity, the invariant is also described as the largest cardinality of a proper stalled subset (Shitov, 2016). The monotonicity underlying the extremal definitions is standard: if $S\subseteq V(G)$3 is a zero forcing set, then any larger set is also zero forcing, and if $S\subseteq V(G)$4 is failed, then any smaller set is also failed (Abara et al., 2022).

The literature distinguishes carefully between maximum and maximal failed sets. A maximum failed zero forcing set has size $S\subseteq V(G)$5, whereas a maximal failed zero forcing set is inclusion-maximal among failed sets (Jacob, 31 Jan 2025). Every maximum failed set is maximal, but not conversely (Jacob, 31 Jan 2025). This distinction becomes central in the theory of well-failed graphs.

2. Stalled sets, forts, and structural duality

A major structural development is the identification of failed zero forcing with fort structure. A fort is a nonempty set $S\subseteq V(G)$6 such that for every vertex $S\subseteq V(G)$7,

$S\subseteq V(G)$8

Equivalently, no vertex outside $S\subseteq V(G)$9 has exactly one neighbor in $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$0 (Jacob, 31 Jan 2025). Forts are exactly the complements of stalled sets (Jacob, 31 Jan 2025).

This yields a precise equivalence between several inclusion-minimal and inclusion-maximal objects. Lemma $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$1 in the well-failed paper states the equivalences

$$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$2

(Jacob, 31 Jan 2025). Consequently, the study of maximal failed sets is exactly the study of minimal forts.

This complementarity underlies the notion of a well-failed graph: a graph $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$3 is well-failed if every maximal failed set is a maximum failed set; equivalently, every minimal fort is minimum (Jacob, 31 Jan 2025). The paper develops this as the failed-zero-forcing analogue of well-covered or well-dominated behavior (Jacob, 31 Jan 2025).

The fort perspective also supports vertex-level irrelevance theory. A vertex is fort-irrelevant if it lies in no minimal fort, zero-forcing-irrelevant if it lies in no minimal zero forcing set, and failed-zero-forcing-irrelevant if it lies in no maximal failed set (Jacob, 31 Jan 2025). A major theorem proves that a vertex is fort-irrelevant if and only if it is zero-forcing-irrelevant (Jacob, 31 Jan 2025). In trees, fort-irrelevant vertices are precisely star centers (Jacob, 31 Jan 2025). The same paper also proves that a vertex is in every minimal fort if and only if it is an end vertex and the graph is a path; equivalently, failed-zero-forcing-irrelevant vertices are exactly end vertices of paths (Jacob, 31 Jan 2025).

This structural theory suggests that failed zero forcing is not merely the complement of zero forcing at the level of optimization, but has its own inclusion-minimal obstruction objects. A plausible implication is that fort-based methods can be more effective than direct forcing arguments when classifying extremal failed sets.

3. Extremal bounds and general inequalities

One of the central global results is a universal lower bound. For every graph $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$4 on $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$5 vertices,

$$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$6

(Ufferman et al., 2022). This answers affirmatively the question whether $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$7 must become arbitrarily large with graph order: for every $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$8, taking $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$9 guarantees $G=(V,E)$0 (Ufferman et al., 2022). The bound is best possible because

$G=(V,E)$1

for paths $G=(V,E)$2 (Ufferman et al., 2022).

The proof of the bound proceeds first for graphs with minimum degree at least $G=(V,E)$3, via a partition $G=(V,E)$4 such that every vertex in $G=(V,E)$5 has at least two neighbors in $G=(V,E)$6 and every vertex in $G=(V,E)$7 has at least two neighbors in $G=(V,E)$8; filling all of $G=(V,E)$9 or all of $v$0 then yields a stalled set (Ufferman et al., 2022). The general theorem follows by induction handling degree-$v$1 and degree-$v$2 vertices via leaf deletion and path condensation (Ufferman et al., 2022).

For standard and positive semidefinite failed zero forcing, the general inequalities

$v$3

are recorded in the minimum-rank study (Abara et al., 2022). The same paper proves

$v$4

for every graph $v$5, since any set stalled under PSD forcing is also stalled under standard forcing (Abara et al., 2022).

Several extremal characterizations are known. For standard failed zero forcing, $v$6 if and only if $v$7 has an isolated vertex, and for connected $v$8, $v$9 if and only if $u$0 has a module of order $u$1 (Gomez et al., 2021, Abara et al., 2022). For PSD failed zero forcing, $u$2 if and only if $u$3 has an isolated vertex, while for connected $u$4,

$u$5

(Abara et al., 2022).

For disconnected graphs, failed zero forcing decomposes by components. If $u$6 are the connected components of $u$7, then

$u$8

(Abara et al., 2022). The same formula holds for $u$9 with $v$0 in place of $v$1 (Abara et al., 2022).

4. Computational complexity

The computational status of failed zero forcing was established by proving NP-completeness of the associated decision problems. The paper “On the complexity of failed zero forcing” formulates:

• FAILED ZERO FORCING: given a finite simple graph $v$2 and an integer $v$3, determine whether $v$4 contains a proper stalled subset of cardinality at least $v$5;
• FAILED SKEW ZERO FORCING: given $v$6 and $v$7, determine whether $v$8 contains a proper skew stalled subset of cardinality at least $v$9

and proves that both are NP-complete (Shitov, 2016). Consequently, the optimization problems of computing the failed zero forcing number and the skew failed forcing number are NP-hard (Shitov, 2016).

The reduction is from INDEPENDENT SET on connected simple graphs (Shitov, 2016). From a graph $u$0 with $u$1, the construction forms a graph $u$2 by subdividing every edge $u$3 with a new vertex $u$4, attaching to each $u$5 a path

$u$6

and adding a vertex $u$7 adjacent to every $u$8 (Shitov, 2016). The resulting graph satisfies

$u$9

The key theorem states: if $u$0 is the largest cardinality of an independent set of $u$1, then the largest proper stalled subset of $u$2 has cardinality

$u$3

and the same conclusion holds for the largest proper skew stalled subset (Shitov, 2016). Hence the reduction

$u$4

is polynomial (Shitov, 2016).

The proof is based on a chain of structural observations. The lower bound comes from extending an independent set $u$5 of the original graph by adding all path vertices $u$6 (Shitov, 2016). The upper bound forces any oversized stalled set to contain all long-path vertices, exclude $u$7, and intersect the original vertex set $u$8 in an independent set (Shitov, 2016). This identifies the failed zero forcing optimization problem as computationally intractable in the standard NP-hard sense.

5. Exact values, classifications, and graph families

A substantial body of work computes $u$9 or $S\subseteq V(G)$0 for specific families.

Standard failed zero forcing

For the graph families studied in the minimum-rank paper, the following values are given (Abara et al., 2022):

Graph family	Value
$S\subseteq V(G)$1	$S\subseteq V(G)$2
$S\subseteq V(G)$3, $S\subseteq V(G)$4	$S\subseteq V(G)$5
$S\subseteq V(G)$6, $S\subseteq V(G)$7	$S\subseteq V(G)$8
$S\subseteq V(G)$9, $S\subseteq V(G)$00	$S\subseteq V(G)$01
$S\subseteq V(G)$02, $S\subseteq V(G)$03	$S\subseteq V(G)$04
$S\subseteq V(G)$05, $S\subseteq V(G)$06	$S\subseteq V(G)$07
$S\subseteq V(G)$08	$S\subseteq V(G)$09
$S\subseteq V(G)$10, $S\subseteq V(G)$11	$S\subseteq V(G)$12
$S\subseteq V(G)$13	$S\subseteq V(G)$14
$S\subseteq V(G)$15, $S\subseteq V(G)$16	$S\subseteq V(G)$17

The same paper records for wheels: $S\subseteq V(G)$18, and for $S\subseteq V(G)$19, $S\subseteq V(G)$20,

$S\subseteq V(G)$21

(Abara et al., 2022).

A separate classification completely determines the graphs with $S\subseteq V(G)$22. There are exactly $S\subseteq V(G)$23 such graphs, all on at most $S\subseteq V(G)$24 vertices; no graph on $S\subseteq V(G)$25 or more vertices has failed zero forcing number $S\subseteq V(G)$26, since every graph with at least $S\subseteq V(G)$27 vertices satisfies $S\subseteq V(G)$28 (Gomez et al., 2021). The disconnected graphs with $S\subseteq V(G)$29 are precisely

$S\subseteq V(G)$30

(Gomez et al., 2021).

The same paper uses structural lemmas involving pendant triangles, cherries, and cut vertices with multiple path components to exclude larger graphs (Gomez et al., 2021). It also proves a monotonicity-type sufficient condition: if $S\subseteq V(G)$31 is a failed zero forcing set in $S\subseteq V(G)$32 and every blue vertex in $S\subseteq V(G)$33 has at least two white neighbors, then $S\subseteq V(G)$34 remains failed in any connected supergraph obtained by adding vertices and edges (Gomez et al., 2021).

Positive semidefinite failed zero forcing

For the PSD variant, exact values include (Abara et al., 2022):

Graph family	Value
$S\subseteq V(G)$35	$S\subseteq V(G)$36
$S\subseteq V(G)$37	$S\subseteq V(G)$38
$S\subseteq V(G)$39, $S\subseteq V(G)$40	$S\subseteq V(G)$41
$S\subseteq V(G)$42	$S\subseteq V(G)$43
$S\subseteq V(G)$44, $S\subseteq V(G)$45	$S\subseteq V(G)$46
$S\subseteq V(G)$47, $S\subseteq V(G)$48	$S\subseteq V(G)$49 if $S\subseteq V(G)$50; $S\subseteq V(G)$51 if $S\subseteq V(G)$52; $S\subseteq V(G)$53 if $S\subseteq V(G)$54
$S\subseteq V(G)$55, $S\subseteq V(G)$56	$S\subseteq V(G)$57
$S\subseteq V(G)$58	$S\subseteq V(G)$59
$S\subseteq V(G)$60, $S\subseteq V(G)$61	$S\subseteq V(G)$62

A major structural theorem states

$S\subseteq V(G)$63

and

$S\subseteq V(G)$64

(Abara et al., 2022). This sharply contrasts with standard failed zero forcing, where paths and many other trees have large $S\subseteq V(G)$65.

Trees, cycles, and well-failed graphs

The well-failed paper characterizes well-failed trees: a tree $S\subseteq V(G)$66 is well-failed if and only if one of the following holds (Jacob, 31 Jan 2025):

1. $S\subseteq V(G)$67 where $S\subseteq V(G)$68 or $S\subseteq V(G)$69,
2. $S\subseteq V(G)$70, the tree obtained by subdividing every edge of $S\subseteq V(G)$71 once,
3. $S\subseteq V(G)$72 is leafy.

A graph is leafy if every non-leaf vertex has a double pendant, i.e. two leaf neighbors sharing that vertex as a common neighbor (Jacob, 31 Jan 2025). Every leafy graph is well-failed (Jacob, 31 Jan 2025). The same paper states that cycles $S\subseteq V(G)$73 are well-failed if and only if $S\subseteq V(G)$74 or $S\subseteq V(G)$75, every complete graph is well-failed, every complete bipartite graph is well-failed, and the Petersen graph is well-failed (Jacob, 31 Jan 2025).

This suggests that uniformity of maximal failed sets is rare but structurally tractable in certain symmetric or highly pendant-rich families.

6. Variants, extensions, and related models

Skew failed zero forcing

In skew zero forcing, a vertex need not be colored in order to force; any vertex may force its unique uncolored neighbor (Johnson et al., 2022). The corresponding failed skew zero forcing number $S\subseteq V(G)$76 is the largest size of a set that does not skew force the entire graph (Johnson et al., 2022). The complexity paper proves that computing the skew failed forcing number is NP-hard via the same reduction used for the standard invariant (Shitov, 2016).

A complete characterization is known for graphs with $S\subseteq V(G)$77. The disconnected case consists only of $S\subseteq V(G)$78; apart from $S\subseteq V(G)$79 and $S\subseteq V(G)$80, the connected graphs are described by a detailed structural theorem built around the existence of exactly one disjoint $S\subseteq V(G)$81-blocking and the absence of other blocking configurations (Johnson et al., 2022). The paper emphasizes a qualitative contrast: changing from standard forcing to skew forcing yields an infinite number of graphs with $S\subseteq V(G)$82 (Johnson et al., 2022).

Directed graphs

Failed zero forcing has also been extended to digraphs. For a digraph $S\subseteq V(G)$83, the color change rule uses unique empty out-neighbors, defining $S\subseteq V(G)$84 as the maximum size of a failed zero forcing set (Adams et al., 2019). The paper proves:

• $S\subseteq V(G)$85 if and only if $S\subseteq V(G)$86 has a source,
• $S\subseteq V(G)$87 if and only if $S\subseteq V(G)$88 is a directed cycle,
• $S\subseteq V(G)$89 if and only if there exist $S\subseteq V(G)$90 with

$S\subseteq V(G)$91

and every vertex has positive indegree (Adams et al., 2019).

Exact formulas are obtained for directed acyclic graphs, weak paths, weak cycles, weakly connected line digraphs, de Bruijn digraphs, and Kautz digraphs (Adams et al., 2019). In particular, for any integers $S\subseteq V(G)$92 and $S\subseteq V(G)$93, there exists a weak cycle $S\subseteq V(G)$94 on $S\subseteq V(G)$95 vertices with $S\subseteq V(G)$96 (Adams et al., 2019).

Product graphs and iterated graph models

For products of graphs, one line of work constructs maximal failed zero forcing sets for Cartesian products, strong products, lexicographic products, and coronas (Abara et al., 2022). The paper proves, among other results,

$S\subseteq V(G)$97

and

$S\subseteq V(G)$98

under the stated parameter ranges (Abara et al., 2022). It also gives lower bounds such as

$S\subseteq V(G)$99

and formulas for lexicographic products and coronas in terms of the failed zero forcing numbers of the factors (Abara et al., 2022).

In iterative deterministic graph models, failed zero forcing shows sharply different behavior in ILT and ILAT graphs. For ILAT graphs, repeated anticloning leads to near-universal behavior: for sufficiently large $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$00, the failed zero forcing number can take only one of four values,

$$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$01

and never $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$02 once $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$03 (Brice et al., 16 Jul 2025). In contrast, ILT behavior remains sensitive to the base graph (Brice et al., 16 Jul 2025).

Relation to minimum rank

The paper on minimum rank and failed zero forcing develops a kernel-based interpretation. If there exists a nonzero vector $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$04 with support disjoint from $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$05, then $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$06 is a failed zero forcing set; under the stated nullity hypothesis, the converse also holds (Abara et al., 2022). The paper proves that for certain graph families,

$$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$07

only in specific small or special cases (Abara et al., 2022). This places failed zero forcing within the broader zero-forcing/minimum-rank program rather than treating it as a purely combinatorial curiosity.

A plausible implication is that failed zero forcing can serve as a complementary extremal certificate: instead of lower-bounding minimum rank through successful forcing, it captures obstruction size through maximal failure.

7. Conceptual significance and neighboring notions

The failed zero forcing number sits at the intersection of propagation processes, obstruction theory, and algorithmic graph invariants. Its significance derives partly from the central role of zero forcing itself in minimum rank, controllability of linear dynamical systems, and quantum control settings (Vazquez, 2020, Trefois et al., 2014). Since failed zero forcing asks for the largest initial set that still cannot complete propagation, it measures a complementary notion of resistance to control.

Several nearby notions clarify what failed zero forcing is not. In the power-law zero-forcing paper, “failed” is used informally to describe regimes where the zero forcing number is near-maximal, i.e. $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$08, meaning that control from a small seed set is ineffective (Vazquez, 2020). That usage is conceptually adjacent but distinct from the graph invariant $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$09, which is defined by maximal unsuccessful initial sets rather than by the size of minimum successful ones.

Likewise, leaky forcing studies robustness under disabled force sources. An $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$10-leaky forcing set must succeed despite $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$11 leaks, and the $$F(G)=\max\{|S|:S\text{ is a failed zero forcing set of }G\}$$12-leaky forcing number is the minimum size of such a set (Alameda et al., 2020). This addresses failure tolerance of successful forcing, not maximal unsuccessful sets. The two perspectives are complementary: failed zero forcing measures how large an unsuccessful set can be, while leaky forcing measures how large a successful set must be under adversarial disruptions.

The current state of the subject combines hardness, exact classification in low-parameter regimes, broad extremal lower bounds, rich variant theory, and increasingly structural approaches through forts and graph families. The resulting picture is that failed zero forcing is a distinct extremal theory within zero forcing: computationally intractable in general (Shitov, 2016), linearly large in every sufficiently large graph (Ufferman et al., 2022), structurally dual to minimal forts (Jacob, 31 Jan 2025), and highly sensitive to the forcing rule and graph model (Abara et al., 2022, Johnson et al., 2022, Brice et al., 16 Jul 2025).