Path eccentricity of $k$-AT-free graphs and application on graphs with the consecutive ones property

2024-03-08

AI Summary

This paper investigates the minimum-eccentricity path problem on connected graphs—i.e., finding a path that minimizes the maximum distance (path eccentricity) to all vertices. Focusing on the class of $k$-AT-free graphs, we establish, for the first time, a tight upper bound of $k$ on path eccentricity, thereby forging an exact structural linkage between graph topology and path eccentricity. Furthermore, we uncover a deep connection between path eccentricity and the consecutive-ones property (C1P): we prove that graphs admitting a diagonal-correctable C1P representation are necessarily 2-AT-free, which improves the known path eccentricity upper bound for this class from 1 to 2—resolving an open problem posed by Gómez and Gutiérrez. Our approach integrates structural characterizations of AT-free graphs, adjacency matrix transformations, and distance-based parametric modeling. These results provide novel theoretical foundations for facility location, network centrality analysis, and graph compression.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• Minimize path eccentricity in connected graphs.
• Study path eccentricity in k-AT-free graphs.
• Link path eccentricity to consecutive ones property.

Innovation

• Minimizes path eccentricity in graphs
• Analyzes $k$-AT-free graphs properties
• Links consecutive ones property to eccentricity