Abstract

The tree breadth tb(G) of a connected graph G is the smallest non-negative integer ρ such that G has a tree decomposition whose bags all have radius at most ρ. We show that, given a connected graph G of order n and size m, one can construct in time O(mlog⁡n) an additive tree O(tb(G)log⁡n)-spanner of G, that is, a spanning subtree T of G in which dT(u,v)≤dG(u,v)+O(tb(G)log⁡n) for every two vertices u and v of G. This improves earlier results of Dragan and Köhler (2014) [8], who obtained a multiplicative error of the same order, and of Dragan and Abu-Ata (2014) [6], who achieved the same additive error with a collection of O(log⁡n) trees.

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