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

TL;DR

The paper addresses the path eccentricity problem by proving a tight bound pe(G) ≤ k for every k-AT-free graph, thereby extending known results from AT-free graphs to the broader k-AT-free class. It also investigates the connection between path eccentricity and the consecutive ones property by introducing the *-C1P, showing that graphs with this property are 2-AT-free and hence have pe(G) ≤ 2, and conjecturing a potential pe(G) ≤ 1 bound. The work combines structural lemmas on induced paths and neighborhood orderings with a minimal counterexample approach to derive the central bounds, and it positions *-C1P as a natural generalization bridging C1P and AT-free graph families. These results advance understanding of how combinatorial matrix properties influence central-path metrics and offer a framework for bounding path eccentricity in broader graph classes with practical implications for facility-location-type problems.

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 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.

Figures (5)

• Figure 1: Summary of the path eccentricity of some classes of graphs.
• Figure 2: Examples of graphs and the associated matrices with ordering function $\mu(v_i)=i, \forall i$.
• Figure 3: Family of graphs having the *-C1P, with ordering function $\mu(v_i)=i, \forall i$.
• Figure 4: Representation of the structure used in the proof of \ref{['thm:kATfree_pek']}.
• Figure 5: A biconvex graph with the asteroidal triple $\{v_1,v_4,v_5\}$.

Theorems & Definitions (19)

• Theorem 1: ATfree
• Theorem 2: PathEcc
• Theorem 3
• Theorem 4
• Corollary 5
• Conjecture 5
• Definition
• Theorem 5
• proof
• Claim 5.1