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arxiv: 2606.22246 · v1 · pith:O533EG6Dnew · submitted 2026-06-20 · 🧮 math.CO

Transmission Zero Forcing

Adam H. Berliner , Chassidy Bozemanand Karen L. Collins , Mary Flagg , Veronika Furst , Mark Hunnell This is my paper

Pith reviewed 2026-06-26 11:28 UTC · model grok-4.3

classification 🧮 math.CO
keywords transmission zero forcingzero forcinggraph parametersiterative processesgraph operationscolor change rule
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The pith

Transmission zero forcing generalizes zero forcing by letting multiple neighbors accumulate weight on a vertex until it exceeds a threshold.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a new graph process in which an initial set of vertices each carries unit weight and transmits additional weight to neighbors according to the standard zero forcing color-change rule at a fixed transmission proportion. A vertex fills once its total weight surpasses the transmission threshold and can then transmit onward. The transmission zero forcing number of a graph is the smallest size of such an initial set that eventually fills every vertex. This construction explicitly permits a vertex to receive forcing contributions from several neighbors rather than from a single neighbor. The authors supply computational tools, evaluate the number on standard families such as paths and cycles, and examine its behavior under common graph operations.

Core claim

Transmission zero forcing begins with a set S of vertices each assigned unit weight. Each filled vertex may increase the weight of an unfilled neighbor at the transmission proportion whenever the zero forcing color-change rule applies. A vertex becomes filled when its weight exceeds the transmission threshold. The transmission zero forcing number is the minimum cardinality of an initial set S that causes every vertex to fill under this rule. The process is a direct generalization of zero forcing in which forcing may be distributed across multiple neighbors.

What carries the argument

The transmission zero forcing process: an iterative weight-accumulation rule driven by a fixed transmission proportion and transmission threshold that extends the zero forcing color-change rule to allow multi-neighbor contributions.

If this is right

  • Exact values of the transmission zero forcing number are obtained for several common graph families.
  • The number behaves predictably under standard graph operations including disjoint union and Cartesian products.
  • The parameter remains distinct from the ordinary zero forcing number on at least some graph classes while recovering it in limiting cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction supplies a tunable family of forcing parameters controlled by two real numbers, allowing direct comparison of different diffusion speeds on the same graph.
  • Because the process is monotonic, standard zero-forcing bounds and algorithms may be adapted by rescaling the threshold.

Load-bearing premise

The transmission proportion and transmission threshold are fixed positive real numbers that make the process well-defined, monotonic, and able to produce a distinct nontrivial graph parameter on ordinary graphs.

What would settle it

An explicit computation, for a concrete graph such as the cycle C_5 or the path P_n, showing either that the process fails to fill the graph from any finite initial set or that the minimum initial-set size cannot be consistently defined once the proportion and threshold are fixed.

Figures

Figures reproduced from arXiv: 2606.22246 by Adam H. Berliner, Chassidy Bozemanand Karen L. Collins, Mark Hunnell, Mary Flagg, Veronika Furst.

Figure 1
Figure 1. Figure 1: The graph G in Example 3.1.2, with the initially filled sets B1 (left), B2 (middle), and B3 (right) shown. Observation 3.1.3. Although it is known that there exists no vertex that must be included in all minimum zero forcing sets of a connected graph [4], the set B3 is the unique minimum p0.8, 0.9q￾transmission forcing set of the graph G in Theorem 3.1.2. In Theorem 3.3.15, we will see one vertex that must… view at source ↗
Figure 2
Figure 2. Figure 2: The graph G in Example 3.1.5, with the initially filled set B shown and final filling achieved in 3 time steps (left) or 4 time steps (right). 3.2 Size of Transmission Zero Forcing Sets We investigate the extreme values of Zα,βpGq, starting with the lower bound. We begin by noting that results for zero forcing can sometimes be directly leveraged in the study of transmission zero forcing. Proposition 3.2.1.… view at source ↗
Figure 3
Figure 3. Figure 3: The graph G in Example 3.3.13. The connection between propagation time for standard zero forcing, propagation time for trans￾mission zero forcing, and the minimum weight of a filled vertex is subtle. The next two examples show us that the propagation time also heavily depends on the choice of α and β. Example 3.3.14. Let G be the graph in [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The graph G in Theorem 3.3.14. Example 3.3.15. Consider the graph G in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The graph G in Theorem 3.3.15 with the initially filled sets B1 (top) and B2 (bottom). 4 Graph Families In this section we consider the transmission zero forcing number of several graph families, drawing on the results for ZpGq and Theorem 3.1.1. 4.1 Complete Graphs and Complete Bipartite Graphs If G is a complete graph, then ZpGq “ n ´ 1. Thus the characterization of Zα,βpGq is a direct consequence of The… view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the graph Gpℓ, kq and a minimum zero forcing set of size ℓ that consists of all the leaves adjacent to v indicated by filled vertices. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The graph G3 pℓ, kq. Observation 5.0.5. For ℓ, k ě 2, we have ZpGs pℓ, kqq “ ℓ ` s ´ 1. This follows because at least ℓ ´ 1 degree-1 neighbors of v and at least s ´ 1 degree-1 neighbors of x have to be included in any zero forcing set, but by themselves, they do not constitute a zero forcing set. However, a set consisting of these vertices with the addition of the last leaf adjacent to x is a zero forcing … view at source ↗
Figure 8
Figure 8. Figure 8: Let B be the set of all degree-1 neighbors of v and the vertex z. Then at time step 1, wpvq “ ℓα, and the transmission forcing process ends when the last vertex on the path receives a weight of ℓαk`1 ě β. So Zα,βpHq ď |B| “ ℓ ` 1. Let e “ vz and note that in the graph H{e, the identified vertex v “ z is adjacent to all other vertices; in particular, it is adjacent to every vertex in P ´ v. An pα, βq-transm… view at source ↗
Figure 8
Figure 8. Figure 8: Illustration of the graph H in Theorem 5.3.2 Acknowledgments The work of M. Flagg is partially supported by NSF award 2331634. The work of V. Furst is partially supported by NSF award DMS-2331072. The work of M. Hunnell is partially supported by NSF award 2447261. This project began at an American Institute of Mathematics (AIM) Workshop. The authors are grateful to AIM and the NSF for their support. We als… view at source ↗
read the original abstract

We initiate the study of transmission zero forcing, a variant of the well-studied zero forcing graph parameter. In this variant, a subset of vertices is assigned an initial unit weight, and these vertices can increase the weight of a neighbor subject to the zero forcing color change rule at a rate determined by the transmission proportion. A vertex is considered filled when its weight exceeds the transmission threshold, at which point the process can continue. The transmission zero forcing number of a graph is the minimum cardinality of the initial set that results in all vertices exceeding the transmission threshold. This iterative graph coloring process is a generalization of zero forcing that allows for a vertex to be forced by multiple neighbors. We develop tools for studying this graph parameter, determine its value on some common classes of graphs, and investigate its behavior under various graph operations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces transmission zero forcing as a generalization of zero forcing. An initial set of vertices is assigned unit weight; these vertices transmit weight to neighbors according to a fixed transmission proportion, and a vertex is filled once its accumulated weight exceeds a fixed transmission threshold. The transmission zero forcing number is the minimum size of an initial set that fills the entire graph. The authors develop tools for the parameter, compute its value on common graph classes, and examine its behavior under graph operations.

Significance. If the definitions, tools, and computed values hold, the work establishes a tunable, additive generalization of zero forcing that permits multiple neighbors to contribute to forcing. This supplies concrete values on standard classes and operational results that can serve as a baseline for further study of weighted propagation processes on graphs.

minor comments (3)
  1. [Abstract / §1] Abstract and §1: the transmission proportion and transmission threshold are described as fixed positive reals, but their precise domains and any restrictions needed to guarantee monotonicity and termination should be stated explicitly in the definition section.
  2. [Abstract] The abstract states that values are determined on 'some common classes of graphs' but does not name them; the introduction or a dedicated section should list the families (paths, cycles, trees, etc.) for which exact values or bounds are obtained.
  3. Notation for the transmission zero forcing number (presumably Z_t(G) or similar) should be introduced once and used consistently; any dependence on the two parameters should be indicated in the notation or stated as fixed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review, positive summary of the transmission zero forcing parameter, and recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces transmission zero forcing via an explicit new definition: an iterative weighted coloring process with fixed transmission proportion and threshold parameters. No load-bearing claim reduces by equation or self-citation to a fitted input, prior result by the same authors, or renaming of a known quantity. Values on graph classes and behavior under operations are computed from the definition itself; the central premise is the introduction and initial exploration of a distinct parameter, which is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 2 invented entities

The central claim rests on the newly introduced concepts of transmission proportion and transmission threshold as modeling choices; these function as free parameters in the definition with no independent evidence supplied in the abstract.

free parameters (2)
  • transmission proportion
    Rate at which weight is transmitted; chosen as part of the model definition.
  • transmission threshold
    Weight level required for a vertex to become filled; chosen as part of the model definition.
axioms (1)
  • standard math The underlying structure is a simple undirected graph.
    Inherited from the zero forcing literature referenced in the abstract.
invented entities (2)
  • transmission proportion no independent evidence
    purpose: Controls the rate of weight increase to neighbors.
    New modeling choice introduced to generalize the forcing rule.
  • transmission threshold no independent evidence
    purpose: Determines when a vertex is considered filled.
    New modeling choice introduced to generalize the forcing rule.

pith-pipeline@v0.9.1-grok · 5674 in / 1341 out tokens · 31831 ms · 2026-06-26T11:28:37.200608+00:00 · methodology

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