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.
Toric ideals generated by quadratic binomials
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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.