Characterization results of all shortest paths interval routing schemes

We give complete characterizations of the classes of graphs with uniform cost links which admit optimum all shortest paths 1 ? SLIRS (Strict Linear Interval Routing Schemes) and 1 ? LIRS (Linear Interval Routing Schemes). The characterization of all the other Interval Routing Schemes with uniform cost links that represent only a single shortest path is known to be NP-Complete. We also show that the class of graphs with dynamic cost links that admit optimum all shortest paths k ? IRS (SIRS, LIRS, SLIRS) are equivalent to the class of graphs with dynamic cost links that admit 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).