Leaky Forcing in Graphs

• Leaky forcing is a robust variant of zero forcing on graphs where an initial blue set must color the entire graph despite up to ℓ vertices being non‐forcing leaks.
• It requires that every non-colored vertex is reachable by at least ℓ+1 distinct blue vertices, ensuring redundancy and fault tolerance in propagation.
• Exact formulas and bounds for leaky forcing numbers have been established for many graph families, providing critical insights into network control and resilience.

Searching arXiv for papers on leaky forcing and closely related variants. Leaky forcing is a fault-tolerant variant of zero forcing on a graph in which an initial blue set must color the entire graph even after an adversary designates up to $\ell$ vertices as leaks, meaning that those vertices do not participate in forcing. The associated parameter, the $\ell$-leaky forcing number $Z_{(\ell)}(G)$, extends the zero forcing number by $Z_{(0)}(G)=Z(G)$ and satisfies the monotone chain $Z_{(0)}(G)\leq Z_{(1)}(G)\leq \cdots$. Introduced to model unreliable junctures in networks, leaky forcing has developed into a framework for robustness in combinatorial propagation, with recursive characterizations, fort-based duality, equivalence among several leak models, connections to strong structural controllability, and exact or sharp results for multiple graph families (Dillman et al., 2019, Alameda et al., 2020, Alameda et al., 2020, Abbas, 2023).

1. Core definitions and forcing mechanism

In the zero forcing game on a simple graph $G=(V,E)$, one begins with an initial blue set $B\subseteq V$. At each step, a blue vertex $v$ can force a neighbor $u$ if $u$ is its only uncolored neighbor. A set that eventually colors every vertex blue is a zero forcing set, and the minimum cardinality of such a set is the zero forcing number $\ell$0. Leaky forcing modifies this rule by allowing up to $\ell$1 leaks to be placed after the initial blue vertices are chosen. A leaky vertex does not participate in forcing; one formulation describes vertices with leaks as behaving as if they are attached to an extra neighbor, making the color change rule stricter. A set $\ell$2 is an $\ell$3-leaky forcing set if, regardless of which $\ell$4 vertices become leaky, the process still colors all vertices, and $\ell$5 denotes the minimum size of such a set (Herrman, 2022).

The same framework is also written in terms of a fixed leak set $\ell$6: the leaky derived set $\ell$7 is the set of vertices eventually colored by zero forcing when vertices in $\ell$8 are not allowed to force. In this notation, $\ell$9 is an $Z_{(\ell)}(G)$0-leaky forcing set precisely when $Z_{(\ell)}(G)$1 for every $Z_{(\ell)}(G)$2 of size $Z_{(\ell)}(G)$3. This formulation is especially useful when leaky forcing is imported into network-control problems, where the initial blue set is interpreted as a leader set or input set (Abbas, 2023).

2. Characterizations, forts, and resilience

A central structural fact is that leaky forcing is governed by redundancy of possible parents in forcing processes. If $Z_{(\ell)}(G)$4 is an $Z_{(\ell)}(G)$5-leaky forcing set, then for every vertex $Z_{(\ell)}(G)$6, there must be at least $Z_{(\ell)}(G)$7 distinct blue vertices that can force $Z_{(\ell)}(G)$8 in possible forcing processes. For $Z_{(\ell)}(G)$9, this becomes an exact characterization: $Z_{(0)}(G)=Z(G)$0 is a $Z_{(0)}(G)=Z(G)$1-leaky forcing set if and only if every $Z_{(0)}(G)=Z(G)$2 can be forced in at least two ways, by two different blue vertices in some possible processes. More generally, $Z_{(0)}(G)=Z(G)$3 is an $Z_{(0)}(G)=Z(G)$4-leaky forcing set if and only if it is an $Z_{(0)}(G)=Z(G)$5-leaky forcing set and, for every set $Z_{(0)}(G)=Z(G)$6 of $Z_{(0)}(G)=Z(G)$7 leaks and every $Z_{(0)}(G)=Z(G)$8, there exist $Z_{(0)}(G)=Z(G)$9 with $Z_{(0)}(G)\leq Z_{(1)}(G)\leq \cdots$0 (Alameda et al., 2020, Alameda et al., 2020).

The theory also admits a fort formulation. In the leaky setting, forts are subsets that can block propagation under adversarial leak placement, and a set is an $Z_{(0)}(G)\leq Z_{(1)}(G)\leq \cdots$1-leaky forcing set if and only if it intersects every $Z_{(0)}(G)\leq Z_{(1)}(G)\leq \cdots$2-leaky fort. This dual viewpoint underlies both exact proofs and optimization formulations. Closely related is the degree obstruction: any $Z_{(0)}(G)\leq Z_{(1)}(G)\leq \cdots$3-leaky forcing set must contain all vertices of degree at most $Z_{(0)}(G)\leq Z_{(1)}(G)\leq \cdots$4, and if $Z_{(0)}(G)\leq Z_{(1)}(G)\leq \cdots$5 then $Z_{(0)}(G)\leq Z_{(1)}(G)\leq \cdots$6 (Dillman et al., 2019, Bjorkman et al., 4 Aug 2025).

Resilience is the special case in which the fault-tolerant parameter does not increase relative to ordinary zero forcing. A graph $Z_{(0)}(G)\leq Z_{(1)}(G)\leq \cdots$7 is called $Z_{(0)}(G)\leq Z_{(1)}(G)\leq \cdots$8-resilient if

$Z_{(0)}(G)\leq Z_{(1)}(G)\leq \cdots$9

This condition imposes immediate structural restrictions. In particular, if $G=(V,E)$0 is $G=(V,E)$1-resilient, then $G=(V,E)$2 (Alameda et al., 2020). A plausible implication is that resilience is best viewed not as a marginal property of a specific forcing chronology but as a global redundancy condition on the set of all admissible chronologies.

3. Equivalent leak models and controllability interpretations

One of the most important generalizations is that the location of the fault can be modeled in several formally different ways without changing the minimum size of a robust forcing set. Besides vertex leaks, an edge leak is an edge $G=(V,E)$3 across which neither $G=(V,E)$4 nor $G=(V,E)$5 is allowed, and a specified leak is a single prohibited direction $G=(V,E)$6. The resulting parameters are the $G=(V,E)$7-edge-leaky forcing number $G=(V,E)$8, the specified $G=(V,E)$9-leaky forcing number $B\subseteq V$0, and the mixed $B\subseteq V$1-leaky forcing number $B\subseteq V$2 when leak types are combined. The main equivalence theorem states that for every graph $B\subseteq V$3 and $B\subseteq V$4,

$B\subseteq V$5

This shows that the worst-case number of disruptions, rather than their type, controls the parameter (Alameda et al., 2020).

An analogous equivalence appears in strong structural controllability. For a graph $B\subseteq V$6, with graph-constrained symmetric matrices $B\subseteq V$7 and dynamics

$B\subseteq V$8

a leader set $B\subseteq V$9 renders the network strong structurally controllable if and only if $v$0 is a zero forcing set. In the resilient setting, three threat models are considered: leaky nodes, non-forcing edges, and removable edges. The main theorem states that for a fixed $v$1 and $v$2, the following are equivalent: $v$3 is an $v$4-LFS, $v$5 is an $v$6-EFS, and $v$7 is an $v$8-FSR. Thus resilience to $v$9 misbehaving nodes, $u$0 non-forcing edges, and $u$1 edge removals coincide at the level of leader selection (Abbas, 2023).

4. Exact values and bounds for major graph families

For several classical graph families, the leaky forcing number is known exactly. For paths,

$u$2

and for cycles,

$u$3

For complete graphs,

$u$4

while for wheel graphs,

$u$5

For trees, if $u$6 is a tree and $u$7 is the set of vertices of degree at most $u$8, then $u$9; in particular, for a tree with $u$0 leaves, excluding $u$1, one has $u$2, and the optimal forcing sets are exactly the leaves (Dillman et al., 2019, Alameda et al., 2020).

For two-dimensional products and related families, the emphasis has been on one-leak robustness. For all $u$3,

$u$4

resolving the question of whether a grid could exceed that upper bound; if $u$5, then $u$6. For supertriangles $u$7, the known values and bounds are

$u$8

These results are typical of the subject: explicit constructions are paired with two-direction forcing arguments that certify robustness to leaks (Alameda et al., 2020).

The hypercube has been a recurrent test case. For the $u$9-dimensional hypercube $\ell$00, it is proved that

$\ell$01

by coloring one $\ell$02-dimensional subcube and then using a combinatorial counting argument on the remaining vertices. Earlier computations showed this bound is tight for $\ell$03, $\ell$04, and $\ell$05, and the conjecture is

$\ell$06

The same paper observes that the result automatically yields upper bounds for $\ell$07 because $\ell$08 for $\ell$09 (Herrman, 2022).

Subsequent work broadened the catalog of exact formulas. For direct products, $\ell$10 and $\ell$11 are both $\ell$12-resilient, with

$\ell$13

and

$\ell$14

The same paper proves

$\ell$15

for all $\ell$16 (Herrman et al., 2024).

For unicyclic graphs, there is now a complete determination of leaky forcing numbers. If $\ell$17 is unicyclic, then for $\ell$18,

$\ell$19

For $\ell$20 and $\ell$21, the values are given by case distinctions depending on girth and on the arrangement of degree-$\ell$22 and degree-$\ell$23 vertices on the unique cycle. For generalized Petersen graphs $\ell$24, the known results include

$\ell$25

the upper bound

$\ell$26

and, for $\ell$27 and $\ell$28,

$\ell$29

The same work also characterizes the connected graphs with extremal $\ell$30-leaky forcing number: $\ell$31 and

$\ell$32

(Bjorkman et al., 4 Aug 2025)

5. Algorithmic methods and sensitivity to graph modifications

Exact computation of leaky forcing numbers is algorithmically difficult. A fort-based integer programming method adapts the standard zero-forcing approach: for each fort $\ell$33, one imposes the covering constraint

$\ell$34

where $\ell$35 indicates whether $\ell$36 is chosen initially. The implementation described for leaky forcing uses constraint generation: solve an initial program, test whether the resulting set fails under some leak placement, add the corresponding fort-like obstruction as a new constraint, and iterate. This framework was used computationally for grids and cubic graphs and made explicit the combinatorial explosion created by possible leak placements (Dillman et al., 2019).

From the perspective of resilient controllability, finding minimal $\ell$37-LFS is NP-hard because minimal zero forcing is NP-hard. For this reason, a greedy algorithm was developed for the $\ell$38-leak setting: initialize with any zero forcing set, repeatedly add the vertex maximizing the size of $\ell$39—the set of non-input nodes with at least two distinct forcers—and optionally prune redundant leaders. Numerical evaluation on Erdős–Rényi and Barabási–Albert graphs reports near-optimal leader sets, a shrinking gap as graphs become denser, and orders-of-magnitude computational savings relative to exhaustive search (Abbas, 2023).

Leaky forcing is also sensitive, but not arbitrarily sensitive, to local graph modification. For any edge $\ell$40,

$\ell$41

and these bounds are tight for all $\ell$42. For any vertex $\ell$43 of degree $\ell$44,

$\ell$45

again with tightness for $\ell$46. At the same time, the early theory emphasized a counterintuitive phenomenon: adding edges can make a graph more resilient to leaks, so monotonicity with respect to edge addition fails in the naive sense one might expect from ordinary forcing heuristics (Bjorkman et al., 4 Aug 2025, Dillman et al., 2019).

6. Variants, recent extensions, and open problems

Leaky forcing has also been extended to positive semidefinite forcing. In $\ell$47-leaky positive semidefinite forcing, leaks are combined with the positive semidefinite color change rule, yielding the parameter $\ell$48. The basic properties parallel the standard theory: $\ell$49 and every $\ell$50-leaky psd forcing set must contain all vertices of degree at most $\ell$51. Exact formulas are known for several families, including

$\ell$52

and the theory introduces $\ell$53-leaky positive semidefinite forts, with the characterization that a set is an $\ell$54-leaky psd forcing set if and only if it intersects every $\ell$55-leaky psd fort (Elias et al., 2023).

A recent exact classification for Hopi rectangle graphs $\ell$56 shows how fort methods can determine an entire leaky forcing sequence. For this family,

$\ell$57

$\ell$58

$\ell$59

and for all $\ell$60,

$\ell$61

The proofs use the induced-subgraph realization inside $\ell$62, explicit forcing chains for $\ell$63, and leaky forts built from the degree-$\ell$64 boundary geometry for larger $\ell$65 (Jr et al., 25 Sep 2025).

Several open problems remain central. The hypercube conjecture

$\ell$66

is known for $\ell$67 but unresolved in general. For generalized Petersen graphs, open directions include determining $\ell$68 and $\ell$69 for general $\ell$70 and $\ell$71. Other structural questions ask whether every graph admits nested minimal leaky forcing sets $\ell$72, and whether every graph has some minimal $\ell$73-leaky forcing set containing a minimum zero forcing set. In product graphs, $\ell$74 is conjectured not to be $\ell$75-resilient, while the $\ell$76-resilience of $\ell$77 remains open; more broadly, no counterexamples are known to the question of whether all Cartesian products $\ell$78 with $\ell$79 are $\ell$80-resilient (Herrman, 2022, Herrman et al., 2024).

Leaky forcing therefore sits at the intersection of zero forcing, robustness, and network control. Its mature form is a family of equivalent adversarial forcing models governed by redundancy of force origins, fort obstructions, and degree constraints, while its current frontier lies in exact classifications for structured graph families, scalable computation, and the separation or unification of its standard and positive semidefinite variants.