  We investigate the Minimum Eccentricity Shortest Path problem in some
structured graph classes. It asks for a given graph to find a shortest path
with minimum eccentricity. Although it is NP-hard in general graphs, we
demonstrate that a minimum eccentricity shortest path can be found in linear
time for distance-hereditary graphs (generalizing the previous result for
trees) and give a generalised approach which allows to solve the problem in
polynomial time for other graph classes. This includes chordal graphs, dually
chordal graphs, graphs with bounded tree-length, and graphs with bounded
hyperbolicity. Additionally, we give a simple algorithm to compute an additive
approximation for graphs with bounded tree-length and graphs with bounded
hyperbolicity.
