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Optimal Symmetry Breaking for Graph Problems

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Abstract

Symmetry breaking is a crucial technique to solve many graph problems. However, current state-of-the-art techniques break graph symmetries only partially, causing search algorithms to unnecessarily explore many isomorphic parts of the search space. We study properties of perfect symmetry breaking for graph problems. One promising and surprising result on small-sized graphs—up to order five—is that perfect symmetry breaking can be achieved using a compact propositional formula in which each literal occurs at most twice. At least for small graphs, perfect symmetry breaking can be expressed more compactly than the existing (partial) symmetry-breaking methods. We present several techniques to compute and analyze perfect symmetry-breaking formulas.

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Notes

  1. The isolators and CNF formulas mentioned in this paper are available at https://github.com/marijnheule/isolator.

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Acknowledgements

The author is supported by the National Science Foundation under Grant No. CCF-1526760 and acknowledges the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing grid resources that have contributed to the research results reported within this paper.

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  1. Department of Computer Science, The University of Texas, Gates Dell Complex, Room 7.714, 2317 Speedway, M/S D9500, Austin, TX, 78712-0233, USA

    Marijn J. H. Heule

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  1. Marijn J. H. Heule
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Correspondence to Marijn J. H. Heule.

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Appendices

Appendix A: Enumeration of Optimal Isolators for Graphs with Four Vertices

$$\begin{aligned}&\quad (\overline{ac}\vee {bc})\wedge (\overline{bc}\vee {ad})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bc}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee \overline{ad}\vee \overline{ab}\vee {ac})\wedge (\overline{bd}\vee {ab}\vee {ac}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {ad})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {bc})\wedge (\overline{bd}\vee \overline{ab}\vee {ad})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{cd}\vee {bd})\wedge (\overline{bc}\vee {ab}\vee {ad})\\&\quad (\overline{ac}\vee {bc})\wedge (\overline{ad}\vee {bc}\vee {bd})\wedge (\overline{ad}\vee \overline{ac}\vee {bd})\wedge (\overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{ab}\vee {ac}\vee {cd})\wedge (\overline{bd}\vee {cd})\wedge (\overline{cd}\vee {ad})\\&\quad (\overline{ac}\vee {bc})\wedge (\overline{ad}\vee {bc}\vee {bd})\wedge (\overline{ac}\vee {bd})\wedge (\overline{bd}\vee {ad})\wedge (\overline{bc}\vee {cd})\wedge (\overline{cd}\vee \overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{cd}\vee {ab}\vee {bd})\\&\quad (\overline{ac}\vee {bc})\wedge (\overline{bc}\vee {ac}\vee {ad})\wedge (\overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{bd}\vee {ad})\wedge (\overline{ad}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {bd})\wedge (\overline{cd}\vee \overline{ac}\vee {bd})\\&\quad (\overline{ac}\vee {bc})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bd}\vee {ad})\wedge (\overline{cd}\vee \overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{bc}\vee \overline{ab}\vee {cd})\wedge (\overline{ad}\vee {ab}\vee {ac}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {bd})\\&\quad (\overline{ac}\vee {bc})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bd}\vee {ad})\wedge (\overline{bc}\vee \overline{ab}\vee {cd})\wedge (\overline{ab}\vee {bd}\vee {cd})\wedge (\overline{cd}\vee \overline{bd}\vee \overline{ab}\vee {ac})\\&\quad (\overline{ac}\vee {bc})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bd}\vee {ad})\wedge (\overline{bc}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee \overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{cd}\vee {ab}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {bc})\wedge (\overline{ad}\vee {bc}\vee {bd})\wedge (\overline{bd}\vee {ad})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bd})\wedge (\overline{ac}\vee {ab}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ad}\vee \overline{ac}\vee {bc})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee {ac}\vee {bc}\vee {cd})\wedge (\overline{bd}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {ad})\wedge (\overline{bc}\vee {ab}\vee {ad})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {bc})\wedge (\overline{ad}\vee \overline{ac}\vee {bd})\wedge (\overline{bd}\vee {ad})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{ad}\vee {bc}\vee {cd})\wedge (\overline{cd}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{bd}\vee {ad})\wedge (\overline{bd}\vee \overline{ac}\vee {bc})\wedge (\overline{ad}\vee {bc}\vee {cd})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bd})\wedge (\overline{ac}\vee {ab}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ad}\vee {bd})\wedge (\overline{ad}\vee \overline{ac}\vee {bc})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{bd}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {ad})\wedge (\overline{ac}\vee {ab}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ad}\vee \overline{ac}\vee {bc})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{bd}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {ad})\wedge (\overline{bc}\vee {ab}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {bc})\wedge (\overline{bc}\vee {ac}\vee {ad})\wedge (\overline{bd}\vee {ad})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{cd}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ad}\vee {ac}\vee {bc})\wedge (\overline{bd}\vee {ad})\wedge (\overline{bd}\vee {bc})\wedge (\overline{bc}\vee \overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bd})\wedge (\overline{ac}\vee {ab}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{bd}\vee {bc})\wedge (\overline{bd}\vee {ad})\wedge (\overline{bc}\vee \overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bd})\wedge (\overline{ac}\vee {ab}\vee {bd})\wedge (\overline{bc}\vee {ab}\vee {ad})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ad}\vee \overline{ac}\vee {bc})\wedge (\overline{bc}\vee \overline{ac}\vee {ad})\wedge (\overline{ad}\vee {bc}\vee {bd})\wedge (\overline{bd}\vee {cd})\wedge (\overline{ac}\vee {cd})\wedge (\overline{cd}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ad}\vee \overline{ac}\vee {bc})\wedge (\overline{bc}\vee \overline{ac}\vee {ad})\wedge (\overline{bc}\vee {ad}\vee {bd})\wedge (\overline{bd}\vee {cd})\wedge (\overline{ac}\vee {cd})\wedge (\overline{cd}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ad}\vee \overline{ac}\vee {bc})\wedge (\overline{bc}\vee \overline{ac}\vee {ad})\wedge (\overline{ac}\vee {bd})\wedge (\overline{ad}\vee {bc}\vee {bd})\wedge (\overline{bd}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {ad}\vee {bc})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ad}\vee \overline{ac}\vee {bc})\wedge (\overline{bc}\vee \overline{ac}\vee {ad})\wedge (\overline{bc}\vee {ad}\vee {bd})\wedge (\overline{ac}\vee {bd})\wedge (\overline{bd}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {ad}\vee {bc})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ad}\vee \overline{ac}\vee {bc})\wedge (\overline{bc}\vee \overline{ac}\vee {ad})\wedge (\overline{ad}\vee {bc}\vee {cd})\wedge (\overline{ac}\vee {cd})\wedge (\overline{cd}\vee {bd})\wedge (\overline{bd}\vee {ab}\vee {ad}\vee {bc})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ad}\vee \overline{ac}\vee {bc})\wedge (\overline{bc}\vee \overline{ac}\vee {ad})\wedge (\overline{ac}\vee {cd})\wedge (\overline{bc}\vee {ad}\vee {cd})\wedge (\overline{cd}\vee {bd})\wedge (\overline{bd}\vee {ab}\vee {ad}\vee {bc})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{ac}\vee {bc})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{ab}\vee {cd})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{bc}\vee {ab}\vee {bd})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{ac}\vee {bc})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{ab}\vee {cd})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {bd})\\&\quad (\overline{ac}\vee {bc})\wedge (\overline{ac}\vee {ad})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{ab}\vee {cd})\wedge (\overline{ad}\vee {ab}\vee {bd})\\&\quad (\overline{ac}\vee {bc})\wedge (\overline{ac}\vee {ad})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{ab}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {bd})\\ \end{aligned}$$
$$\begin{aligned}&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {ad})\wedge (\overline{ac}\vee {bc})\wedge (\overline{ad}\vee {bc}\vee {bd})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {ad})\wedge (\overline{ac}\vee {bc})\wedge (\overline{bc}\vee {ad}\vee {bd})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bd})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{ac}\vee {bc})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{ab}\vee {cd})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {ad}\vee {bc})\\&\quad (\overline{ac}\vee {bc})\wedge (\overline{ac}\vee {ad})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{ab}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {ad}\vee {bc})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{ad}\vee {bc})\wedge (\overline{bc}\vee {bd})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{ab}\vee {bc}\vee {cd})\wedge (\overline{cd}\vee \overline{bc}\vee \overline{ab}\vee {ac})\wedge (\overline{bd}\vee {ab}\vee {ac}\vee {cd})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{ad}\vee {bc})\wedge (\overline{bc}\vee {bd})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee \overline{bc}\vee \overline{ab}\vee {ac})\wedge (\overline{bd}\vee {ab}\vee {ac}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {bc})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{ad}\vee {bc})\wedge (\overline{bc}\vee {bd})\wedge (\overline{ab}\vee {bc}\vee {cd})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee \overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{bd}\vee {ab}\vee {ad}\vee {cd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {ad})\wedge (\overline{bd}\vee \overline{ac}\vee {bc})\wedge (\overline{bc}\vee \overline{ab}\vee {cd})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{cd}\vee {bd})\wedge (\overline{ad}\vee {ab}\vee {bc})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{bc}\vee \overline{ac}\vee {bd})\wedge (\overline{bc}\vee {ad}\vee {bd})\wedge (\overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{ab}\vee {ac}\vee {cd})\wedge (\overline{bd}\vee {cd})\wedge (\overline{cd}\vee {bc})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{ad}\vee {ac}\vee {bc})\wedge (\overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{bd}\vee {bc})\wedge (\overline{bc}\vee {cd})\wedge (\overline{cd}\vee \overline{ac}\vee {bd})\wedge (\overline{cd}\vee {ab}\vee {bd})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{bc}\vee {ad}\vee {bd})\wedge (\overline{ac}\vee {bd})\wedge (\overline{bd}\vee {bc})\wedge (\overline{ad}\vee {cd})\wedge (\overline{cd}\vee \overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{cd}\vee {ab}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {ad})\wedge (\overline{bc}\vee \overline{ac}\vee {bd})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{bd}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bc})\wedge (\overline{ad}\vee {ab}\vee {bc})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee {bc})\wedge (\overline{ab}\vee {bd}\vee {cd})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee \overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{bc}\vee {ab}\vee {ac}\vee {cd})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee {bc})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee \overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{bc}\vee {ab}\vee {ac}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {bd})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee {bc})\wedge (\overline{ab}\vee {bd}\vee {cd})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee \overline{bc}\vee \overline{ab}\vee {ac})\wedge (\overline{bc}\vee {ab}\vee {ad}\vee {cd})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{bc}\vee {bd})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee {bc})\wedge (\overline{ab}\vee {bc}\vee {cd})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee \overline{bd}\vee \overline{ab}\vee {ac})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{bc}\vee {bd})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee {bc})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee \overline{bd}\vee \overline{ab}\vee {ac})\wedge (\overline{cd}\vee {ab}\vee {bc})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{bc}\vee \overline{ac}\vee {ad})\wedge (\overline{ac}\vee {ad}\vee {bd})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bd}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bc})\wedge (\overline{ad}\vee {ab}\vee {bc})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {ad})\wedge (\overline{ad}\vee {ac}\vee {bc})\wedge (\overline{bd}\vee {bc})\wedge (\overline{bc}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bd})\wedge (\overline{ac}\vee {ab}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {ad})\wedge (\overline{bc}\vee {ad}\vee {bd})\wedge (\overline{bd}\vee {bc})\wedge (\overline{bc}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bd})\wedge (\overline{ac}\vee {ab}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{bc}\vee \overline{ac}\vee {ad})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bd}\vee {ac}\vee {ad}\vee {cd})\wedge (\overline{bd}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bc})\wedge (\overline{ad}\vee {ab}\vee {bc})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {ad})\wedge (\overline{bc}\vee \overline{ac}\vee {bd})\wedge (\overline{bd}\vee {bc})\wedge (\overline{bc}\vee {ad}\vee {cd})\wedge (\overline{bd}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{bd}\vee {bc})\wedge (\overline{bc}\vee {ad}\vee {cd})\wedge (\overline{bc}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bd})\wedge (\overline{cd}\vee \overline{ac}\vee {ad})\wedge (\overline{ac}\vee {ab}\vee {bd})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{bd}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee \overline{ac}\vee {ad})\wedge (\overline{cd}\vee {bc})\wedge (\overline{ad}\vee {ab}\vee {bc})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {ad})\wedge (\overline{bd}\vee {bc})\wedge (\overline{bc}\vee \overline{ab}\vee {cd})\wedge (\overline{bd}\vee \overline{ad}\vee {ac}\vee {cd})\wedge (\overline{cd}\vee {bd})\wedge (\overline{ad}\vee {ab}\vee {bc})\\&\quad (\overline{ab}\vee {ac})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bc}\vee \overline{ac}\vee {ad})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{bd}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {bc})\wedge (\overline{ac}\vee {ab}\vee {bd})\\ \end{aligned}$$
$$\begin{aligned}&\quad (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {ad})\wedge (\overline{ad}\vee {ac}\vee {bc})\wedge (\overline{bd}\vee {bc})\wedge (\overline{bc}\vee \overline{ab}\vee {cd})\wedge (\overline{bd}\vee {ac}\vee {cd})\wedge (\overline{cd}\vee {bd})\\&\quad (\overline{ad}\vee {ac})\wedge (\overline{bc}\vee \overline{ac}\vee {ad})\wedge (\overline{ac}\vee {ad}\vee {bd})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bd}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {bc})\wedge (\overline{cd}\vee \overline{ad}\vee {bc})\\&\quad (\overline{ad}\vee {ac})\wedge (\overline{ad}\vee {bc})\wedge (\overline{bc}\vee \overline{ac}\vee {ad})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bd}\vee {ac}\vee {bc})\wedge (\overline{ac}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {bc})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{ad}\vee {ac})\wedge (\overline{ad}\vee {bc})\wedge (\overline{bc}\vee {bd})\wedge (\overline{ab}\vee {bc}\vee {cd})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{bd}\vee {ab}\vee {bc})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{ad}\vee {ac})\wedge (\overline{ac}\vee {bc})\wedge (\overline{bc}\vee {bd})\wedge (\overline{bd}\vee {bc}\vee {cd})\wedge (\overline{ac}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {bc})\\&\quad (\overline{ac}\vee {ad})\wedge (\overline{bc}\vee {bd})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee {bc})\wedge (\overline{ad}\vee \overline{ab}\vee {cd})\wedge (\overline{ab}\vee {bc}\vee {cd})\wedge (\overline{cd}\vee \overline{ad}\vee {ac})\\&\quad (\overline{ad}\vee \overline{ac}\vee {bc})\wedge (\overline{bc}\vee {ac})\wedge (\overline{ad}\vee {bd})\wedge (\overline{ac}\vee {bc}\vee {bd})\wedge (\overline{bd}\vee {cd})\wedge (\overline{cd}\vee \overline{bc}\vee {ad})\wedge (\overline{cd}\vee {ab}\vee {ad})\\&\quad (\overline{ad}\vee \overline{ac}\vee {bc})\wedge (\overline{bc}\vee {ad})\wedge (\overline{bc}\vee {ac})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee {ac}\vee {ad})\wedge (\overline{ac}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {ad})\\&\quad (\overline{ac}\vee {bc})\wedge (\overline{bc}\vee {ac})\wedge (\overline{bc}\vee {ad})\wedge (\overline{ad}\vee {bd})\wedge (\overline{bd}\vee {ad}\vee {cd})\wedge (\overline{ac}\vee \overline{ab}\vee {cd})\wedge (\overline{cd}\vee {ab}\vee {ad})\\ \end{aligned}$$

Appendix B: Enumeration of Optimal Isolators for Graphs with Five Vertices

$$\begin{aligned} P_{5{\mathrm{a}}}:= & {} (\overline{ab}\vee {de})\wedge ({ac}\vee \overline{bc})\wedge (\overline{ac}\vee {cd})\wedge ({ae}\vee \overline{de})\wedge (\overline{ae}\vee {bd})\wedge (\overline{bd}\vee {be})\wedge ({ac}\vee \overline{cd}\vee {ce})\wedge \phantom {x}\\&({ad}\vee \overline{be}\vee {cd})\wedge ({ad}\vee \overline{be}\vee \overline{ce})\wedge ({ae}\vee {bc}\vee \overline{ce})\wedge (\overline{ae}\vee \overline{bc}\vee {ce})\wedge (\overline{ac}\vee \overline{ad}\vee {bc}\vee \overline{bd})\\ P_{5{\mathrm{b}}}:= & {} (\overline{ab}\vee {de})\wedge ({ac}\vee \overline{bc})\wedge (\overline{ac}\vee {cd})\wedge ({ae}\vee \overline{bd})\wedge (\overline{ae}\vee {be})\wedge ({bd}\vee \overline{de})\wedge ({ac}\vee {ad}\vee \overline{be})\wedge \phantom {x}\\&({ac}\vee {bd}\vee \overline{cd})\wedge ({ad}\vee \overline{bc}\vee \overline{be})\wedge ({ae}\vee {bc}\vee \overline{ce})\wedge (\overline{ac}\vee {bc}\vee \overline{bd}\vee \overline{ce})\wedge (\overline{ad}\vee \overline{ae}\vee \overline{cd}\vee {ce})\\ P_{5{\mathrm{c}}}:= & {} (\overline{ab}\vee {de})\wedge (\overline{ac}\vee {bc})\wedge ({ad}\vee \overline{de})\wedge (\overline{ad}\vee {be})\wedge ({ae}\vee \overline{be})\wedge (\overline{bc}\vee {cd})\wedge ({ac}\vee \overline{bc}\vee \overline{de})\wedge \phantom {x}\\&({ac}\vee {be}\vee \overline{ce})\wedge (\overline{ac}\vee \overline{ae}\vee {bd})\wedge ({ad}\vee {bc}\vee \overline{cd})\wedge (\overline{ae}\vee {bc}\vee {bd})\wedge (\overline{bd}\vee \overline{be}\vee \overline{cd}\vee {ce})\\ P_{5{\mathrm{d}}}:= & {} (\overline{ac}\vee {bc})\wedge ({ad}\vee \overline{de})\wedge (\overline{ad}\vee {be})\wedge ({ae}\vee \overline{be})\wedge (\overline{ae}\vee {bd})\wedge (\overline{bc}\vee {cd})\wedge (\overline{ab}\vee \overline{bc}\vee {de})\wedge \phantom {x}\\&(\overline{ab}\vee {ce}\vee {de})\wedge ({ac}\vee \overline{bc}\vee \overline{ce})\wedge ({ae}\vee {bc}\vee \overline{ce})\wedge (\overline{be}\vee \overline{cd}\vee {ce})\wedge ({ab}\vee {ad}\vee {bc}\vee \overline{cd})\\ P_{5{\mathrm{e}}}:= & {} ({ac}\vee \overline{bc})\wedge (\overline{ac}\vee {bc})\wedge ({ad}\vee \overline{ae})\wedge (\overline{ad}\vee {be})\wedge ({ae}\vee \overline{bd})\wedge (\overline{cd}\vee {ce})\wedge ({ab}\vee {bd}\vee \overline{be})\wedge \phantom {x}\\&({ad}\vee {bc}\vee \overline{ce})\wedge (\overline{ad}\vee {bd}\vee \overline{de})\wedge (\overline{bc}\vee \overline{be}\vee {cd})\wedge (\overline{ab}\vee \overline{ac}\vee \overline{cd}\vee {de})\wedge (\overline{ab}\vee {bc}\vee {ce}\vee {de})\\ P_{5{\mathrm{f}}}:= & {} ({ac}\vee \overline{bc})\wedge (\overline{ac}\vee {bc})\wedge ({ad}\vee \overline{bd})\wedge (\overline{ad}\vee {be})\wedge (\overline{ae}\vee {bd})\wedge ({cd}\vee \overline{ce})\wedge ({ab}\vee {ae}\vee \overline{be})\wedge \phantom {x}\\&(\overline{ac}\vee \overline{be}\vee {ce})\wedge ({ad}\vee {bc}\vee \overline{cd})\wedge (\overline{ad}\vee {ae}\vee \overline{de})\wedge (\overline{ab}\vee \overline{ac}\vee \overline{ce}\vee {de})\wedge (\overline{ab}\vee {bc}\vee {cd}\vee {de})\\ P_{5{\mathrm{g}}}:= & {} (\overline{ab}\vee {de})\wedge (\overline{ac}\vee {bc})\wedge (\overline{ac}\vee {cd})\wedge ({ad}\vee \overline{be})\wedge (\overline{ae}\vee {bd})\wedge (\overline{bd}\vee {be})\wedge (\overline{cd}\vee {ce})\wedge \phantom {x}\\&({ab}\vee {cd}\vee \overline{ce})\wedge (\overline{ad}\vee {ae}\vee \overline{cd})\wedge (\overline{ad}\vee {bd}\vee {ce})\wedge (\overline{ab}\vee {ac}\vee \overline{ae}\vee \overline{ce})\wedge ({ae}\vee {bc}\vee {cd}\vee \overline{de})\\ P_{5{\mathrm{h}}}:= & {} (\overline{ab}\vee {de})\wedge (\overline{ac}\vee {bc})\wedge (\overline{ac}\vee {cd})\wedge ({ad}\vee \overline{bd})\wedge (\overline{ae}\vee {be})\wedge ({bd}\vee \overline{be})\wedge (\overline{cd}\vee {ce})\wedge \phantom {x}\\&({ab}\vee {cd}\vee \overline{ce})\wedge (\overline{ad}\vee {ae}\vee {bc})\wedge (\overline{ad}\vee {ae}\vee \overline{cd})\wedge ({be}\vee {ce}\vee \overline{de})\wedge (\overline{ab}\vee {ac}\vee \overline{ae}\vee \overline{ce})\\ \end{aligned}$$
$$\begin{aligned} P_{5{\mathrm{i}}}:= & {} (\overline{ab}\vee {de})\wedge (\overline{ac}\vee {bc})\wedge (\overline{ac}\vee {cd})\wedge ({ad}\vee \overline{bd})\wedge (\overline{ae}\vee {be})\wedge ({bd}\vee \overline{be})\wedge (\overline{cd}\vee {ce})\wedge \phantom {x}\\&({ab}\vee {cd}\vee \overline{ce})\wedge (\overline{ad}\vee {ae}\vee \overline{cd})\wedge (\overline{ad}\vee {be}\vee {ce})\wedge (\overline{ab}\vee {ac}\vee \overline{ae}\vee \overline{ce})\wedge ({ae}\vee {bc}\vee {cd}\vee \overline{de})\\ P_{5{\mathrm{j}}}:= & {} ({ad}\vee \overline{de})\wedge (\overline{ad}\vee {be})\wedge ({ae}\vee \overline{bd})\wedge (\overline{bc}\vee {cd})\wedge ({bd}\vee \overline{be})\wedge (\overline{cd}\vee {ce})\wedge ({ab}\vee {ac}\vee {ad}\vee \overline{ce})\wedge \phantom {x}\\&(\overline{ab}\vee \overline{bc}\vee {de})\wedge (\overline{ab}\vee {cd}\vee {de})\wedge (\overline{ac}\vee {ce}\vee {de})\wedge (\overline{ab}\vee {ac}\vee \overline{be}\vee \overline{ce})\wedge (\overline{ac}\vee {bc}\vee \overline{be}\vee \overline{ce})\\ P_{5{\mathrm{k}}}:= & {} ({ad}\vee \overline{be})\wedge (\overline{ad}\vee {bd})\wedge ({ae}\vee \overline{bd})\wedge (\overline{bc}\vee {cd})\wedge ({be}\vee \overline{de})\wedge (\overline{cd}\vee {ce})\wedge ({ab}\vee {ac}\vee {be}\vee \overline{ce})\wedge \phantom {x}\\&(\overline{ab}\vee \overline{bc}\vee {de})\wedge (\overline{ab}\vee {cd}\vee {de})\wedge (\overline{ac}\vee {ce}\vee {de})\wedge (\overline{ab}\vee {ac}\vee \overline{ad}\vee \overline{ce})\wedge (\overline{ac}\vee \overline{ad}\vee {bc}\vee \overline{ce})\\ P_{5{\mathrm{l}}}:= & {} (\overline{ab}\vee {de})\wedge (\overline{ac}\vee {bc})\wedge ({ad}\vee \overline{ae})\wedge (\overline{ad}\vee {be})\wedge ({ae}\vee \overline{bd})\wedge ({cd}\vee \overline{ce})\wedge ({ab}\vee {ac}\vee \overline{be}\vee {ce})\wedge \phantom {x}\\&(\overline{ac}\vee \overline{ae}\vee {ce})\wedge ({ad}\vee \overline{bc}\vee {cd})\wedge (\overline{ab}\vee {ac}\vee \overline{be}\vee \overline{cd})\wedge ({ac}\vee {bd}\vee \overline{cd}\vee {de})\wedge (\overline{bc}\vee {bd}\vee \overline{cd}\vee \overline{de})\\ P_{5{\mathrm{m}}}:= & {} (\overline{ab}\vee {ac})\wedge (\overline{ac}\vee {ad})\wedge (\overline{bc}\vee {de})\wedge ({bd}\vee \overline{de})\wedge (\overline{bd}\vee {ce})\wedge ({be}\vee \overline{ce})\wedge ({ab}\vee \overline{ae}\vee {ce})\wedge \phantom {x}\\&(\overline{ab}\vee \overline{be}\vee {cd})\wedge ({ac}\vee \overline{ad}\vee {bd})\wedge ({ac}\vee \overline{be}\vee {cd})\wedge (\overline{ac}\vee \overline{ae}\vee {bc}\vee \overline{bd})\wedge (\overline{ad}\vee {ae}\vee \overline{cd}\vee \overline{ce})\\ P_{5{\mathrm{n}}}:= & {} (\overline{ab}\vee {ac})\wedge (\overline{ad}\vee {ae})\wedge ({bd}\vee \overline{ce})\wedge (\overline{bd}\vee {cd})\wedge ({be}\vee \overline{cd})\wedge (\overline{be}\vee {de})\wedge ({ab}\vee \overline{ac}\vee {cd})\wedge \phantom {x}\\&({ac}\vee \overline{ae}\vee {bd})\wedge (\overline{ac}\vee {ad}\vee \overline{bd})\wedge ({ab}\vee {ad}\vee {bc}\vee \overline{de})\wedge (\overline{ab}\vee \overline{bc}\vee \overline{be}\vee {ce})\wedge (\overline{ac}\vee \overline{ad}\vee {bc}\vee \overline{de})\\ P_{5{\mathrm{o}}}:= & {} (\overline{ac}\vee {ad})\wedge (\overline{ad}\vee {ae})\wedge (\overline{bc}\vee {de})\wedge ({bd}\vee \overline{ce})\wedge (\overline{bd}\vee {cd})\wedge (\overline{be}\vee {ce})\wedge (\overline{ab}\vee {ac}\vee \overline{be})\wedge \phantom {x}\\&(\overline{ab}\vee {ac}\vee {ce})\wedge (\overline{ac}\vee {bc}\vee \overline{de})\wedge ({be}\vee \overline{cd}\vee \overline{de})\wedge ({ab}\vee {ad}\vee \overline{ae}\vee {bc})\wedge ({ab}\vee {ad}\vee {be}\vee \overline{cd})\\ P_{5{\mathrm{p}}}:= & {} ({ac}\vee \overline{ad})\wedge (\overline{ac}\vee {ae})\wedge ({bc}\vee \overline{be})\wedge (\overline{bc}\vee {de})\wedge ({bd}\vee \overline{de})\wedge ({be}\vee \overline{cd})\wedge ({ab}\vee {ac}\vee \overline{ae})\wedge \phantom {x}\\&(\overline{ab}\vee {ad}\vee {bc})\wedge (\overline{ab}\vee \overline{bd}\vee {ce})\wedge (\overline{ad}\vee \overline{bc}\vee {cd})\wedge ({ae}\vee \overline{bd}\vee {ce})\wedge (\overline{ac}\vee {ad}\vee \overline{ce}\vee \overline{de})\\ \end{aligned}$$

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Heule, M.J.H. Optimal Symmetry Breaking for Graph Problems. Math.Comput.Sci. 13, 533–548 (2019). https://doi.org/10.1007/s11786-019-00397-5

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