Collective Tree Spanners in Graphs with Bounded Genus, Chordality, Tree-Width, or Clique-Width

Abstract

In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G=(V,E) admits a system of μ collective additive tree r -spanners if there is a system $\mathcal{T}(G)$ of at most μ spanning trees of G such that for any two vertices x,y of G a spanning tree $T \in \mathcal{T}(G)$ exists such that d _T (x,y)≤ d _G (x,y)+r. We describe a general method for constructing a ”small” system of collective additive tree r-spanners with small values of r for ”well” decomposable graphs, and as a byproduct show (among other results) that any weighted planar graph admits a system of $O(\sqrt{n})$ collective additive tree 0–spanners, any weighted graph with tree-width at most k–1 admits a system of k log₂ n collective additive tree 0–spanners, any weighted graph with clique-width at most k admits a system of k log_3/2 n collective additive tree (2w)–spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log₂ n collective additive tree $(2\lfloor{c/2}\rfloor{\sf w})$–spanners and a system of 4log₂ n collective additive tree $(2(\lfloor{c/3}\rfloor{+1}){\sf w})$–spanners (here, w is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4 log₂ n collective additive tree (2w)-spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs.