  Let $G$ be a connected graph. The eccentricity of a path $P$, denoted by
ecc$_G(P)$, is the maximum distance from $P$ to any vertex in $G$. In the
\textsc{Central path} (CP) problem our aim is to find a path of minimum
eccentricity. This problem was introduced by Cockayne et al., in 1981, in the
study of different centrality measures on graphs. They showed that CP can be
solved in linear time in trees, but it is known to be NP-hard in many classes
of graphs such as chordal bipartite graphs, planar 3-connected graphs, split
graphs, etc.
  We investigate the path eccentricity of a connected graph~$G$ as a parameter.
Let pe$(G)$ denote the value of ecc$_G(P)$ for a central path $P$ of $G$. We
obtain tight upper bounds for pe$(G)$ in some graph classes. We show that
pe$(G) \leq 1$ on biconvex graphs and that pe$(G) \leq 2$ on bipartite convex
graphs. Moreover, we design algorithms that find such a path in linear time. On
the other hand, by investigating the longest paths of a graph, we obtain tight
upper bounds for pe$(G)$ on general graphs and $k$-connected graphs.
  Finally, we study the relation between a central path and a longest path in a
graph. We show that on trees, and bipartite permutation graphs, a longest path
is also a central path. Furthermore, for superclasses of these graphs, we
exhibit counterexamples for this property.
