Algorithm and references¶

Pivoter¶

Pivoter is an exact clique counting algorithm due to Shweta Jain and C. Seshadhri, described in:

• The Power of Pivoting for Exact Clique Counting (January 2020).

It avoids listing every maximal clique by organizing computation around a Succinct Clique Tree (SCT) and pivoting ideas adapted from classical clique enumeration, with degeneracy-based structure exploited for sparse graphs.

Reference code from the authors:

• GitHub — sjain12/Pivoter
• Bitbucket mirror

pivoterpy reimplements that workflow with a Python kernel and a Rust extension; it is not a line-for-line port of the authors’ Julia/C code, but follows the same family of ideas.

Degeneracy ordering¶

pivoterpy uses a degeneracy ordering of vertices (Batagelj–Završnik style in the current Rust pipeline) as part of the search structure. That choice affects performance and recursion shape, not the final counts for a fixed simple graph.

Related work (short timeline)¶

These are classic touchpoints in maximal clique listing and clique counting; Pivoter sits in the “exact counting with careful pivoting / structure” line.

April 1971 — Bron–Kerbosch

• C. Bron and J. Kerbosch — foundational backtracking for cliques.
• ACM Algorithm 457 (PDF)

October 2006 — Tomita et al.

• Worst-case analysis and experiments on maximal clique generation.
• Paper (Stanford CS224W readings)

2010–2011 — Eppstein, Löffler, Strash

• Near-optimal listing in sparse graphs and large sparse instances.
• arXiv:1006.5440 · arXiv:1103.0318
• Practical code: quick-cliques

Other Pivoter implementations¶

• Pivoter (Julia) — charunupara/Pivoter
• PyPivoter (Cython) — rckormos/PyPivoter
• pivoterpy (this project) — Python + Rust

Citation¶

Cite the original Pivoter paper for the algorithm.

If you also want to cite this package:

@software{anderson_pivoterpy_2026,
  author       = {Spencer Anderson},
  title        = {{pivoterpy}},
  year         = {2026},
  version      = {2.1.1},
  url          = {https://github.com/7e6i/pivoterpy}
}