Path eccentricity of graphs

TL;DR

This paper studies the path eccentricity in graphs, providing bounds, algorithms for specific classes, and exploring the relationship between longest and central paths.

Contribution

It introduces tight bounds for path eccentricity in various graph classes and develops linear-time algorithms for finding such paths.

Findings

pe(G) ≤ 1 on biconvex graphs

pe(G) ≤ 2 on bipartite convex graphs

Longest paths are central in trees and bipartite permutation graphs

Abstract

Let $G$ be a connected graph. The eccentricity of a path $P$, denoted by ecc$_G(P)$, is the maximum distance from $P$ to any vertex in $G$. In the \textsc{Central path} (CP) problem our aim is to find a path of minimum eccentricity. This problem was introduced by Cockayne et al., in 1981, in the study of different centrality measures on graphs. They showed that CP can be solved in linear time in trees, but it is known to be NP-hard in many classes of graphs such as chordal bipartite graphs, planar 3-connected graphs, split graphs, etc. We investigate the path eccentricity of a connected graph~$G$ as a parameter. Let pe$(G)$ denote the value of ecc$_G(P)$ for a central path $P$ of $G$. We obtain tight upper bounds for pe$(G)$ in some graph classes. We show that pe$(G) \leq 1$ on biconvex graphs and that pe$(G) \leq 2$ on bipartite convex graphs. Moreover, we design algorithms that find such a path in linear time. On the other hand, by investigating the longest paths of a graph, we obtain tight upper bounds for pe$(G)$ on general graphs and $k$-connected graphs. Finally, we study the relation between a central path and a longest path in a graph. We show that on trees, and bipartite permutation graphs, a longest path is also a central path. Furthermore, for superclasses of these graphs, we exhibit counterexamples for this property.