For complete-intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they distance-reduce the circuits, and this property is preserved across homogeneous ideals sharing the same bouquet structure and signature.
Hypergraph encodings of arbi- trary toric ideals.Journal of Combinatorial Theory, Series A, 166:11–41, 2019
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Distance Reduction in Bouquet Decompositions and Toric Ideals of Graphs
For complete-intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they distance-reduce the circuits, and this property is preserved across homogeneous ideals sharing the same bouquet structure and signature.