Temporal zero forcing

Research archive for Zero forcing on temporal graphs: Layer-local and footprint-constrained dynamics by Krishna Harish (Elkins High School, Missouri City, Texas, United States).

The project develops two synchronous finite-lifetime variants of the standard zero-forcing color-change rule:

• Layer-local temporal zero forcing (LL-TZF): uniqueness is evaluated in the active temporal layer.
• Footprint-constrained temporal zero forcing (FC-TZF): uniqueness is evaluated in the aggregate footprint, while the forcing edge must be active in the current layer.

The proved results include sharp growth bounds, exact values for alternating paths and even cycles, aggregation and time-reversal separations, restricted NP-completeness and W[2]-hardness, and fixed-parameter algorithms. The claim-level prior-art assessment is deliberately calibrated: no exact prior formulation of both invariants was found, but the definitions combine mature zero-forcing and temporal-graph ideas. See notes/novelty-audit.md.

Reproduce the computational checks

Requirements: Python 3.9 or newer and no third-party packages. The archived release was tested with Python 3.13.7.

python3 src/verify.py

Expected output:

Verified 127 formula instances and 4,096 exhaustive two-layer schedules; wrote .../results/verification.csv

The command regenerates results/verification.csv. Its expected SHA-256 digest is 7035cd92c06c06b6fa2b1562d476a89ec541589162dad0811e6fedd5a70cfd45.

Archive contents

• paper/main.tex and paper/output/pdf/temporal-zero-forcing.pdf: manuscript source and rendered PDF.
• src/verify.py: standard-library verification program.
• results/verification.csv: 127 exact formula checks and optimum witnesses.
• results/verification_summary.json: pass status and counts for the assertion-only exhaustive checks.
• supplement/S1_Code_and_results.zip: separately uploadable supporting-information archive.
• notes/novelty-audit.md: claim-level novelty audit, closest prior work, and search limits.
• CITATION.cff and .zenodo.json: citation and archival metadata.

Citation

Use the citation metadata supplied by GitHub or Zenodo for release v1.0.0. The version-specific DOI is 10.5281/zenodo.21347118. The all-versions concept DOI is 10.5281/zenodo.21347117.

Licensing

The verification software and associated documentation are available under the MIT License. The manuscript is available under CC BY 4.0.