Collective Tree Spanners and Routing in AT-free Related Graphs

Abstract

In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in AT-free graphs. 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 d_T(x,y) ≤ d_G(x,y)+r. Among other results, we show that AT-free graphs have a system of two collective additive tree 2-spanners (whereas there are trapezoid graphs that do not admit any additive tree 2-spanner). Furthermore, based on this collection, we derive a compact and efficient routing scheme. Also, any DSP-graph (there exists a dominating shortest path) admits an additive tree 4-spanner, a system of two collective additive tree 3-spanners and a system of five collective additive tree 2-spanners.