Theta-ring graphs, IO-compatibility and Delta-matroids

In this paper we introduce O-compatible and IO-compatible graphs. These graphs are related with Delta-matroid representations. We prove that theta-ring graphs are IO compatible. Recall G is a theta-ring graph if and only if the toric ideal P-GO 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 IO-compatible. Recall that the Truemper configurations are the obstructions for the family of theta-ring graphs.