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Antony Halim
[WSS20] Finding Conservation Laws in (Hyper)graph Rewritings
Antony Halim, Minerva Schools at Keck Graduate Institute
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7
Bipartite & Acyclic & Loop : 0.0762
Bipartite & Simple & Loop : 0.0777
Bipartite & Simple & Acyclic : 0.0762
Connected & Loop & Simple : 0.0026
Acyclic & Simple & Loop : 0.0762
Bipartite & Acyclic & Loop & Simple : 0.762
O
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Planar & Acyclic & Loop : 0.0762
Planar & Acyclic & Simple : 0.0762
Planar & Simple & Loop : 0.0777
Acyclic & Loop & Simple : 0.0762
Planar & Acyclic & Loop & Simple : 0.0762
O
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Planar & Bipartite & Loop Free : 0.0762
Planar & Bipartite & Simple : 0.0762
Planar & Loop Free & Simple : 0.0762
Bipartite & Loop Free & Simple : 0.0762
Planar & Simple & Loop Free & Bipartite : 0.0762
O
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Planar & Bipartite & Acyclic : 0.0762
Planar & Bipartite & Simple : 0.0777
Planar & Connected & Simple : 0.0026
Planar & Acyclic & Simple : 0.0762
Bipartite & Acyclic & Simple : 0.0762
O
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7
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w
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P
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6
2
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6
7
Planar & Bipartite & Acyclic : 0.0762
Planar & Bipartite & Loop Free : 0.0777
Planar & Connected & Loop Free : 0.0026
Planar & Acyclic & Loop Free 0.0762
Bipartite & Acyclic & Loop Free : 0.0762
Planar & Acyclic & Loop Free & Bipartite : 0.0762
Conclusions & Recommendations
By analyzing which of the previous combinations of properties can lead to a graph that preserves Connectedness, and that does not get split into many subgraphs due to successive rule applications, we find that hypergraphs and rules combinations that give graphs that are both Hamiltonian and free of Self-Loops have always given outputs that preserve Connectedness. While the graphs that were only Hamiltonian had significantly high rates of preserving Connectedness, only the combination of the two properties, the graph being Hamiltonian and free of Self-Loops, solely had graphs that conserved Connectedness. It is important to note as well that many of the outputs that are preserving Connectedness converged to a fixed state after only one step, but some of the cases, like the one shown in the animation, did not show any convergence within the first twenty steps, and these are expected to be the intriguing cases to consider. It can also be noticed that these cases that preserve Connectedness do not all conserve the properties of the graph being Hamiltonian or having no Self-Loops.
The next step is to use a systematic approach to enumerate all hypergraphs of a similar signature then apply each of the
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rules to each of the enumerated hypergraphs to make sure that there are no cases that give graphs that are both Hamiltonian and Self-Loop free while failing to preserve Connectedness. It is expected that the rules that preserve Connectedness and that do not converge quickly are the ones that can lead to interesting results.
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I would like to thank my Mentor Matthew Szudzik, Stephen Wolfram, and the rest of the WSS staff for making this opportunity possible. Also, I would like to thank Professor Kiel Howe for his continuous help and support.
POSTED BY:
Antony Halim
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