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

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 cliquewidth, 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 T (G) of at most μ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT (x, y) ≤ dG(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( √ 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 log3/2 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 c/2 w)–spanners and a system of 4 log2 n collective additive tree (2( c/3 +1)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.