Easy computation of eccentricity approximating trees

A spanning tree T of a graph G=(V,E) is called eccentricity k-approximating if we have eccT(v)≤eccG(v)+k for every v∈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. (2017). 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.