### Collective Tree Spanners of Graphs

01/01/2006In this paper we introduce a new notion of *collective tree spanners*. We say that a graph *G=(V,E)admits a system of $\mu$ collective additive tree r-spanners* if there is a system *T(G)* of at most $\mu$ spanning trees of *G* such that for any two vertices *x,y* of *G* a spanning tree *T\in \cT(G)* exists such that *d_T(x,y)\leq 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. These results are complemented by lower bounds, which say that any system of collective additive tree 1-spanners must have $\Omega(\sqrt{n})$ spanning trees for some chordal graphs and $\Omega(n)$ spanning trees for some chordal bipartite graphs and some cocomparability graphs. Furthermore, we show that any *c*-chordal graph admits a system of at most log2n collective additive tree (2\lfloor c/2\rfloor)-spanners, any circular-arc graph admits a system of two collective additive tree 2-spanners. Towards establishing these results, we present a general property for graphs, called *(\al,r)$-decomposition*, and show that any $(\al,r)$-decomposable graph *G* with *n* vertices admits a system of at most $\log_{1/\al} n$ 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. For any graph on *n* vertices admitting a system of at most $\mu$ collective additive tree *r*-spanners, there is a routing scheme of deviation *r* with addresses and routing tables of size $O(\mu \log^2n/\log \log n)$ bits per vertex. This leads, for example, to a routing scheme of deviation $(2\lfloor c/2\rfloor)$ with addresses and routing tables of size $O(\log^3n/\log \log n)$ bits per vertex on the class of *c*-chordal graphs.