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.

翻译：中心路径问题是单设施选址问题的一个变体。其目标是在给定连通图$G$中寻找一条路径$P$，使得其离心率（即从$P$到图$G$中任意顶点的最大距离）最小化。图$G$的路径离心率是所有路径中可实现的最小离心率。本文考虑$k$-AT-free图类的路径离心率。在这类图中，任意三个顶点中必存在一对顶点，使得它们之间的每条路径都至少包含第三个顶点闭$k$-邻域中的一个顶点。我们证明这类图的路径离心率以$k$为界。此外，我们回答了Gómez和Gutiérrez提出的问题：路径离心率与连续1性质之间是否存在关联？后者是指二元矩阵存在一个行排列使得每一列中的1连续出现。已知邻接矩阵具有连续1性质的图的路径离心率至多为1，且当增广邻接矩阵（对角线元素为1）具有连续1性质时该结论同样成立。我们将这些结果推广如下：研究通过更改对角线上的某些值后能使邻接矩阵满足连续1性质的图，通过证明这些图是2-AT-free图，从而表明其路径离心率至多为2。