On compact and efficient routing in certain graph classes

Abstract

In this paper we refine the notion of tree-decomposition by introducing acyclic ( R , D ) -clustering, where clusters are subsets of vertices of a graph and R and D are the maximum radius and the maximum diameter of these subsets. We design a routing scheme for graphs admitting induced acyclic ( R , D ) -clustering where the induced radius and the induced diameter of each cluster are at most 2. We show that, by constructing a family of special spanning trees, one can achieve a routing scheme of deviation Δ ⩽ 2 R with labels of size O ( log 3 n / log log n ) bits per vertex and O ( 1 ) routing protocol for these graphs. We investigate also some special graph classes admitting induced acyclic ( R , D ) -clustering with induced radius and diameter less than or equal to 2, namely, chordal bipartite, homogeneously orderable, and interval graphs. We achieve the deviation Δ = 1 for interval graphs and Δ = 2 for chordal bipartite and homogeneously orderable graphs.