  The central path problem is a variation on the single facility location
problem. The aim is to find, in a given connected graph $G$, a path $P$
minimizing its eccentricity, which is the maximal distance from $P$ to any
vertex of the graph $G$. The path eccentricity of $G$ is the minimal
eccentricity achievable over all paths in $G$. In this article we consider the
path eccentricity of the class of the $k$-AT-free graphs. They are graphs in
which any set of three vertices contains a pair for which every path between
them uses at least one vertex of the closed neighborhood at distance $k$ of the
third. We prove that they have path eccentricity bounded by $k$.
  Moreover, we answer a question of G\'omez and Guti\'errez asking if there is
a relation between path eccentricity and the consecutive ones property. The
latter is the property for a binary matrix to admit a permutation of the rows
placing the 1's consecutively on the columns. It was already known that graphs
whose adjacency matrices have the consecutive ones property have path
eccentricity at most 1, and that the same remains true when the augmented
adjacency matrices (with ones on the diagonal) has the consecutive ones
property. We generalize these results as follow. We study graphs whose
adjacency matrices can be made to satisfy the consecutive ones property after
changing some values on the diagonal, and show that those graphs have path
eccentricity at most 2, by showing that they are 2-AT-free.
