Navigating in a Graph by Aid of Its Spanning Tree Metric
SIAM Journal on Discrete Mathematics
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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 clique-width, 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 $\mathcal{T}(G)$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. 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 $O(\sqrt{n})$collective additive tree 0-spanners, any weighted graph with tree-width at most k−1 admits a system of klog 2 n collective additive tree 0-spanners, any weighted graph with clique-width at most k admits a system of klog 3/2 n collective additive tree $(2\mathsf{w})$-spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log 2 n collective additive tree $(2\lfloor c/2\rfloor\mathsf{w})$-spanners and a system of 4log 2 n collective additive tree $(2(\lfloor c/3\rfloor +1)\mathsf {w})$-spanners (here, $\mathsf{w}$is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4log 2 n collective additive tree $(2\mathsf{w})$-spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs.