On the interval routing of chordal rings

The Shortest-Path Interval Routing Scheme is an efficient strategy to code distributed routing algorithms in a compact way. Characterising networks which admit shortest path Strict Interval Routing Scheme using one interval per edge (1-SIRS) is known to be NP-complete. We study 1-SIRS for a popular class of networks, known as chordal rings. We prove that for any chordal ring of degree 4 with a chord of length k(>3), there exists an infinite set of n, such that the chordal ring <1, k>_n (of size n) does not admit a 1-SIRS, regardless of the node labeling. This gives a negative answer to an open question from C. Gavoille (1997) and extends the characterisation of node labelings which allow a shortest path 1-SIRS in chordal rings. Mainly, we study the natural cyclic node labeling and derive an alternative node labeling using isomorphic properties. First we show that there is an infinite class of chordal rings with chord k of even length that do not have a shortest path 1-SIRS. Second, we show the limitation of the cyclic node labeling and its alternative representation when k is odd. Finally, we conjecture that any chordal ring with shortest path 1-SIRS has a node labeling isomorphic to a chordal ring with a cyclic node labeling