Easy computation of eccentricity approximating trees
Résumé
A spanning tree $T$ of a graph $G = (V, E)$ is called eccentricity $k$-approximating if we have $ecc_T (v) \leq ecc_G (v) + k$ for every $v \in V$. Let $ets(G)$ be the minimum $k$ such that $G$ admits an eccentricity $k$-approximating spanning tree. As our main contribution in this paper, we prove that $ets(G)$ can be computed in $O(nm)$-time along with a corresponding spanning tree. This answers an open question of [Dragan et al., DAM'17]. Moreover we also prove that for some classes of graphs such as chordal graphs and hyperbolic graphs, one can compute an eccentricity $O(ets(G))$-approximating spanning tree in quasi linear time. Our proofs are based on simple relationships between eccentricity approximating trees and shortest-path trees.
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