Skew Forcing Irredundance in Graph Theory

• The paper establishes that skew forcing irredundance uses private skew forts to uniquely certify each vertex in a forcing set, linking it to classical irredundance theories.
• It employs a universal blocking-set formulation to derive interlacing inequalities between lower and upper skew forcing irredundance numbers in graphs.
• The study highlights the distinct structure of skew forcing irredundance, noting its component consistency, non-robustness, and contrast with zero forcing and related irredundance parameters.

Searching arXiv for the specified paper and closely related work on skew forcing, irredundance, and TAR reconfiguration. Skew forcing irredundance is the irredundance theory obtained by applying a universal blocking-set construction to skew forcing. In this formulation, the relevant obstructions are skew forts, and a vertex set is skew-forcing irredundant when every one of its vertices is uniquely certified by a private skew fort. The subject was developed as a specialization of a broader theory of irredundance for $X$-set parameters, which places skew forcing irredundance alongside domination irredundance, zero forcing irredundance, PSD forcing irredundance, and vertex cover irredundance, while also isolating a major structural difference: unlike several of those related theories, the skew version is component consistent but not robust (Curtis et al., 23 Sep 2025).

1. Universal blocking-set formulation

The general framework begins with a super $X$-set parameter $X$ and a family of $X$-blocking sets. For a graph $G$, a subset $R\subseteq V(G)$ is an $X$-blocking set if $V(G)\setminus R$ is not an $X$-set. Minimal $X$-blocking sets are those minimal under inclusion. Given a blocking family $X$0, a set $X$1 is $X$2-irredundant if each $X$3 has a private blocking set $X$4 such that

$X$5

The associated irredundance parameter $X$6 is the maximum size of a maximal $X$7-irredundant set. In this theory, minimal blocking sets are the decisive objects: $X$8 is $X$9-irredundant if and only if $X$0 (Curtis et al., 23 Sep 2025).

This framework is designed to recover classical irredundance constructions from the obstruction family appropriate to the parameter under study. Its principal structural consequence is an interlacing theorem: for a component consistent super $X$1-set parameter and an $X$2-irredundant blocking family, the irredundance parameter lies between the original parameter and its upper analogue. In the skew setting, that general statement becomes the inequality chain relating lower skew forcing irredundance, skew forcing, upper skew forcing, and upper skew forcing irredundance. A plausible implication is that skew forcing irredundance should be viewed not as an ad hoc variant, but as the canonical irredundance theory attached to skew forcing once its obstruction family is identified (Curtis et al., 23 Sep 2025).

2. Skew forcing and private skew forts

The skew color change rule allows any vertex $X$3 to force a white vertex $X$4 whenever $X$5 is the only white neighbor of $X$6, that is,

$X$7

A set $X$8 is a skew zero forcing set if repeated application of this rule turns all vertices blue. The skew zero forcing number is denoted $X$9 in one account and $X$0 in another (Curtis et al., 2024, Hogben et al., 2015).

For skew forcing irredundance, the blocking sets are skew forts. A nonempty set $X$1 is a skew fort if

$X$2

The foundational skew-fort theorem states that $X$3 is a skew forcing set if and only if $X$4 for every skew fort of $X$5. Accordingly, the natural skew blocking family is

$X$6

A set is then skew-forcing irredundant exactly when each of its vertices has a private skew fort, i.e.,

$X$7

for some skew fort $X$8 depending on $X$9 (Curtis et al., 23 Sep 2025).

The lower skew forcing irredundance number is the minimum size of a maximal skew-forcing irredundant set, and the upper skew forcing irredundance number is the maximum size of such a set. This definition is directly parallel to zero forcing irredundance, where forts replace closed neighborhoods as the witnesses of irredundance. The main difference is that skew forcing uses skew forts rather than ordinary forts, because the underlying forcing rule allows white vertices to force (Curtis et al., 23 Sep 2025).

3. Parameter interlacing and extremal structure

The universal interlacing theorem specializes to skew forcing irredundance: the lower skew irredundance number is at most the skew forcing number, which is at most the upper skew forcing number, which is at most the upper skew irredundance number. The same source emphasizes that all four skew parameters are distinct in general and supplies examples exhibiting noncomparability phenomena (Curtis et al., 23 Sep 2025).

Several extremal cases are explicit. If the skew forcing number is $G$0, then all skew irredundance parameters are $G$1. For complete graphs $G$2,

$G$3

and the skew irredundance parameters are also $G$4. More generally, for a connected graph $G$5 of order $G$6, the condition that either skew irredundance extremum equals $G$7 is equivalent to a structural partition

$G$8

where $G$9 and $R\subseteq V(G)$0 are nonempty independent sets and

$R\subseteq V(G)$1

This is the skew counterpart of the known structure theorem for graphs with skew forcing number $R\subseteq V(G)$2 (Curtis et al., 23 Sep 2025).

The lower regime behaves differently from the original skew forcing number. The lower skew irredundance number is not comparable in general with skew forcing itself: a path of even order has skew forcing number $R\subseteq V(G)$3 and lower skew irredundance number $R\subseteq V(G)$4, whereas a star has skew forcing number $R\subseteq V(G)$5 and lower skew irredundance number $R\subseteq V(G)$6. This separates skew forcing irredundance from the more tightly aligned behavior familiar from some domination-theoretic inequalities (Curtis et al., 23 Sep 2025).

4. Closure operators, robustness, and TAR reconfiguration

A second universal formulation is given through closure operators. For skew forcing, the closure operator is the final coloring obtained from an initial blue set $R\subseteq V(G)$7: $R\subseteq V(G)$8 This operator is a closure operator, and the associated closure family is component consistent but not inclusive. The blocking family induced by that closure operator is exactly the family of skew forts,

$R\subseteq V(G)$9

Thus the closure-operator approach and the direct fort-based approach coincide in the skew case (Curtis et al., 23 Sep 2025).

The decisive structural issue is robustness. When $X$0, there are no skew forts at all, so the only skew-forcing irredundant set is $X$1. Consequently the skew irredundance parameter is not a robust $X$2-set parameter. This excludes it from the general TAR isomorphism theorem for irredundance parameters that requires robustness together with inclusiveness and component consistency. By contrast, the forcing parameter itself behaves differently: skew forcing is treated as a robust $X$3-set parameter in the TAR framework, and if $X$4 for graphs with no isolated vertices, then $X$5 and $X$6 have the same order and, after relabeling, exactly the same skew forcing sets (Curtis et al., 23 Sep 2025, Curtis et al., 2024).

A common source of confusion is therefore the distinction between the forcing parameter and its irredundance analogue. Skew forcing admits the robust $X$7-set TAR theory, whereas skew forcing irredundance does not inherit the full robust $X$8-set TAR reconstruction theorem because its blocking family fails inclusiveness precisely in the presence of graphs with skew forcing number $X$9 (Curtis et al., 23 Sep 2025).

5. Relation to zero forcing irredundance and adjacent irredundance theories

Zero forcing irredundance was introduced earlier using private forts. In that setting, a fort is a nonempty set $V(G)\setminus R$0 such that

$V(G)\setminus R$1

and a set $V(G)\setminus R$2 is a ZIr-set if every element of $V(G)\setminus R$3 has a private fort. The corresponding extremal parameters are

$V(G)\setminus R$4

They satisfy the chain

$V(G)\setminus R$5

Every skew fort is a standard fort, so every skew-irredundant set is also a standard zero forcing irredundant set. In particular, the upper skew forcing irredundance number is at most $V(G)\setminus R$6, and the inequality can be strict (Curtis et al., 2024, Curtis et al., 23 Sep 2025).

The universal theory in which skew forcing irredundance is embedded also covers PSD forcing irredundance and vertex cover irredundance. For PSD forcing irredundance, the analogous inequality chain holds and the parameter is robust, so the TAR isomorphism theorem applies. For vertex cover irredundance, the blocking sets are edges, the paper proves

$V(G)\setminus R$7

and also

$V(G)\setminus R$8

with $V(G)\setminus R$9 robust. In the domination setting, the classical chain

$X$0

is extended by incorporating upper vertex cover irredundance, not skew forcing irredundance. This explicit exclusion reflects the non-robustness of the skew theory on the $X$1-set side (Curtis et al., 23 Sep 2025).

6. Related skew forcing structure and neighboring obstruction theories

Independent work on skew forcing supplies structural facts that clarify the behavior of skew forcing irredundance. A three-color reformulation of skew zero forcing uses dark blue, light blue, and white vertices, and proves

$X$2

In an optimal skew zero forcing set for a connected graph, every light blue vertex must perform a force before it is forced, no two light blue vertices can be adjacent, and hence the light blue set is independent. The same work gives a leaf-stripping characterization: $X$3 This identifies exactly the graphs on which skew forcing irredundance collapses, because the absence of skew forts when $X$4 is the mechanism behind non-robustness (Hogben et al., 2015).

A neighboring but distinct theory is failed skew zero forcing, whose parameter $X$5 is the largest size of a set that does not force all vertices under the skew rule. The corresponding stalled-set language is not the same as irredundance, but both frameworks are obstruction-based and both are sensitive to the special role of skew propagation. The characterization of graphs with $X$6 shows that the skew rule produces an infinite family governed by $X$7-blockings, odd cycles, even paths, and the exclusion of pendant even cycles and degree-$X$8 vertices. This suggests that obstruction families for skew dynamics are substantially richer than their standard zero forcing counterparts, even when the parameter under study is not itself an irredundance number (Johnson et al., 2022).

Taken together, these results place skew forcing irredundance at the intersection of two developments. One is the universal irredundance program, in which blocking sets generate canonical lower and upper irredundance parameters. The other is the broader study of skew forcing, whose permissive forcing rule creates both the fort-based theory of private witnesses and the exceptional $X$9 regime that prevents a full robust $X$0-set TAR theory (Curtis et al., 23 Sep 2025).