Interval routing on k-trees

A graph has an optimal I-interval routing scheme if it is possible to direct messages along shortest paths by labeling each edge with at most l painwise-disjoint subintervals of the cyclic interval [1...n] (where each node of the graph is labeled by an integer in the range). Although much progress has been made for l = 1, there is as yet no general tight characterization of the classes of graphs associated with larger l. Bodlaender et al. have shown that under the assumption of dynamic cost links, each graph with an optimal l-interval routing scheme has treewidth of at most 4l. For the setting without dynamic cost links, this paper addresses the complementary question of the number of intervals required to label classes of graphs of treewidth k. Although it has been shown that there exist graphs of treewidth 2 that require a nonconstant number of intervals, our work demonstrates a class of graphs of treewidth 2, namely 2-trees, that are guaranteed to allow 3-interval routing schemes. In contrast, this paper presents a 2-tree that cannot have a 2-interval routing scheme. For general k, any k-tree is shown to have an optimal interval routing scheme using 2(k+1) intervals per edge. (C) 1998 Academic Press.