Theta-ring graphs, I 𝒪 -compatibility and Δ -matroids

In this paper we introduce 𝒪 -compatible and I 𝒪 -compatible graphs. These graphs are related with Δ -matroid representations. We prove that theta-ring graphs are I 𝒪 -compatible. Recall G is a theta-ring graph if and only if the toric ideal P_G_𝒪 is a binomial complete intersection for each edge orientation 𝒪 of G . We characterize the 𝒪 -compatible theta-ring graphs. Finally, we characterize which Truemper configurations are 𝒪 -compatible or I 𝒪 -compatible. Recall that the Truemper configurations are the obstructions for the family of theta-ring graphs.