Enumerating Diametral and Radial Vertices and Computing Diameter and Radius of a Graph

Abstract

In this chapter we show an example of enumeration problem whose number of solutions is polynomial and for which a polynomial algorithm exists, that is enumerating diametral vertices, i.e. vertices whose eccentricity is the diameter, and radial vertices, i.e. vertices whose eccentricity is the radius. Intuitively they correspond to periphery and center of a network. After an overview on the centrality measures used in the analysis of biological networks, we show an efficient algorithm to list such vertices. The contribution of this work is not just limited to biological networks, but is even more useful for complex huge network analysis in general. Indeed, even if the complexity of our algorithms is theoretically $O(nm)$, like the text-book algorithm, we show that in practice it runs often in $O(m)$ time. This study implies the analysis of the diameter and radius of a network so that we will evaluate the effectiveness of our algorithms in finding such measures.