Collective Tree Spanners of Graphs

Abstract

In this paper we introduce a new notion of collective tree spanners. 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. Among other results, we show that any chordal graph, chordal bipartite graph or cocomparability graph admits a system of at most log₂ n collective additive tree 2–spanners and any c-chordal graph admits a system of at most log₂ n collective additive tree (2$\lfloor c/2\rfloor$)–spanners. Towards establishing these results, we present a general property for graphs, called (α,r)–decomposition, and show that any (α,r)–decomposable graph G with n vertices admits a system of at most log_1/α n collective additive tree 2r–spanners. We discuss also an application of the collective tree spanners to the problem of designing compact and efficient routing schemes in graphs.