On the Eccentricity Function in Graphs

Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)

Abstract

Given a graph \(G=(V,E)\), the eccentricity of a vertex u is the distance from u to a vertex farthest from u. The set of vertices that minimizes the maximum distance to every other vertex (has minimum eccentricity) constitutes the center of the graph. The minimum eccentricity value represents the graph’s radius. The eccentricity function of a graph can be unimodal or non-unimodal. A graph with unimodal eccentricity function has the property that the eccentricity of every vertex equals its distance to the center plus the radius. A graph with non-unimodal eccentricity function lacks this property. In this work, we characterize each type of eccentricity function and study the impact of each type on the intersection of shortest paths among distant vertex pairs with the center. A shortest path intersects the center if it includes at least one vertex that belongs to the center. In particular, we show that if the eccentricity function is unimodal, all shortest paths among distant vertex pairs intersect the graph’s center. We also discuss when those paths do not intersect the center in graphs with non-unimodal eccentricity functions.

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceKent State UniversityKentUSA

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