Interval Routing Schemes for Circular-Arc Graphs

TL;DR

This paper demonstrates that circular-arc graphs admit a shortest path strict 2-interval routing scheme, solving an open problem and providing an efficient O(n^2) algorithm for constructing such schemes.

Contribution

It proves that circular-arc graphs have strict compactness 2 and presents an outline algorithm to compute the routing scheme efficiently.

Findings

Circular-arc graphs allow strict 2-interval routing schemes for shortest paths.

The constructed scheme can be converted into a 1-interval scheme with minimal additional intervals.

An O(n^2) algorithm is outlined for computing the routing scheme for circular-arc graphs.

Abstract

Interval routing is a space efficient method to realize a distributed routing function. In this paper we show that every circular-arc graph allows a shortest path strict 2-interval routing scheme, i.e., by introducing a global order on the vertices and assigning at most two (strict) intervals in this order to the ends of every edge allows to depict a routing function that implies exclusively shortest paths. Since circular-arc graphs do not allow shortest path 1-interval routing schemes in general, the result implies that the class of circular-arc graphs has strict compactness 2, which was a hitherto open question. Additionally, we show that the constructed 2-interval routing scheme is a 1-interval routing scheme with at most one additional interval assigned at each vertex and we an outline algorithm to calculate the routing scheme for circular-arc graphs in O(n^2) time, where n is the number of vertices.