$k$-AT-free图的路径离心率及其在具有连续1性质的图上的应用
Path eccentricity of $k$-AT-free graphs and application on graphs with the consecutive ones property
AI总结：
本文研究$k$-AT-free图的路径离心率，证明其上界为$k$，并推广了路径离心率与连续1性质的关联结论，指出邻接矩阵修改对角元后满足连续1性质的图路径离心率至多为2。
摘要：
中心路径问题是单设施选址问题的一个变体。其目标是在给定的连通图$G$中找到一条路径$P$，使其离心率最小，其中离心率定义为从$P$到图$G$任意顶点的最大距离。图$G$的路径离心率是$G$中所有路径所能达到的最小离心率。本文研究了$k$-AT-free图类的路径离心率，这类图的定义是：任意三个顶点构成的集合中，都存在一对顶点，使得它们之间的每条路径都至少用到第三个顶点的距离为$k$的闭邻域中的一个顶点。我们证明了这类图的路径离心率以$k$为上界。\n此外，我们回答了Gómez和Gutiérrez提出的一个问题，即路径离心率与连续1性质（consecutive ones property）之间是否存在关联。连续1性质是指二元矩阵存在一种行置换，使得每一列中的1都连续排列。此前已知，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（对角元为1）具有连续1性质时，该结论同样成立。我们对这些结果进行了如下推广：我们研究了邻接矩阵在修改对角线上的部分元素后可满足连续1性质的图，通过证明这类图是2-AT-free图，得出它们的路径离心率至多为2的结论。
英文摘要：
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 Gómez and Gutiérrez 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.
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