$\Delta$-graphs of polytopes in Bruns and Gubeladze $K$-theory

W. Bruns and J. Gubeladze introduced a new variant of algebraic $K$-theory, where \linebreak $K$-groups are additionally parametrized by polytopes of some type. In this paper we propose a notion of stable $E$-equivalence which can be used to calculate $K$-groups for high-dimensional polytopes. Polytopes which are stable $E$-equivalent have similar inner structures and isomorphic $K$-groups. In addition, for each polytope we define a $\Delta$-graph which is an oriented graph being invariant under a stable $E$-equivalence.