We give complete characterizations for the classes of graphs with uniform cost links that admit optimum all shortest paths 1-SLIRS (strict linear interval routing schemes) and 1-LIRS (linear interval routing schemes). The characterization of all the interval routing schemes with uniform cost links that represent only a single shortest path is known to be NP-complete. For any integer k > 0, we also show that the class of graphs with dynamic cost links that admit optimum all shortest paths k-IRS (SIRS, LIRS, SLIRS) is equivalent to the class of graphs with dynamic cost links that admit an optimum single shortest path k-IRS (SIRS, LIRS, SLIRS) and also equivalent to the class of graphs with dynamic cost links that admit single paths up to any constant stretch factor k-IRS (SIRS, LIRS, SLIRS).
