Extended formulations for the integer hull of strictly $\mathrm{\Delta }$-modular cographic polyhedral cones

Conforti et al. give a compact extended formulation for a class of bimodular-constrained integer programs, namely those that model the stable set polytope of a graph with no disjoint odd cycles. We extend their techniques to design compact extended formulations for the integer hull of translated polyhedral cones whose constraint matrix is strictly $\mathrm{\Delta }$-modular and has rows that represent a cographic matroid. Our work generalizes the important special case from Conforti et al. concerning $4$-connected graphs with odd cycle transversal number at least $4$. We also discuss the necessity of our assumptions.