Totally $\mathrm{\Delta }$-Modular Tree Decompositions of Graphic Matrices for Integer Programming

We introduce the tree-decomposition-based parameter totally $\mathrm{\Delta }$-modular treewidth (TDM-treewidth) for matrices with two nonzero entries per row. We show how to solve integer programs whose matrices have bounded TDM-treewidth in polynomial time when variables have bounded domain. This extends previous graph-based decomposition parameters for matrices with at most two nonzero entries per row to include matrices with entries outside of $\{-1,0,1\}$. We also give an analogue of the Grid Theorem of Robertson and Seymour for matrices of bounded TDM-treewidth in the language of rooted signed graphs.