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., 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.