Distance Reduction in Bouquet Decompositions and Toric Ideals of Graphs

Abstract

The distance-reduction property for a generating set, i.e., a Markov basis, of a toric ideal is a condition that ensures tight connectivity of its fibres. In this paper, we study the distance-reduction property for toric ideals of graphs and move on to explore the relationship between the distance-reduction property and the bouquet structure of homogeneous toric ideals, which includes the class of toric ideals of graphs. For toric ideals of graphs which are complete intersection, we show that the minimal Markov bases are distance-reducing if and only if they distance-reduce the circuits of the ideal. We then consider how the distance-reduction properties interact with the bouquet structure of the toric ideal. Bouquets are a combinatorial structure that capture the essential combinatorial information of the toric ideal. Under the condition of homogeneity, we show that, for toric ideals with the same bouquet structure and signature, the distance-reduction properties are preserved. For homogeneous toric ideals whose bouquet matrix is a monomial curve in $\mathbb{A}^3$, we give necessary and sufficient conditions for when the minimal Markov bases are distance-reducing.