Collective Tree Spanners of Graphs

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 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. 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 c/2 )–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 log1/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.