Computation of diameter, radius and center of permutation graphs

Abstract

For a connected graph [Formula: see text], we use the notation [Formula: see text] to represent the distance between two node points [Formula: see text] and [Formula: see text] and it is the minimum of the lengths of all paths between them. The eccentricity [Formula: see text] of a node point [Formula: see text] is considered as the maximum length of all shortest paths starts from [Formula: see text] to the remaining nodes, i.e., [Formula: see text]. The diameter of a graph [Formula: see text], we denote it by [Formula: see text] and it is the length of the longest shortest path in [Formula: see text], i.e., [Formula: see text]. Also, the radius of a graph [Formula: see text], we denote it by the symbol [Formula: see text] and it is the least eccentricity of all node points in [Formula: see text], i.e., [Formula: see text]. The central vertex/node point [Formula: see text] of a graph [Formula: see text] is a node whose eccentricity is same as [Formula: see text]’s radius, i.e., [Formula: see text]. The collection of all central nodes of a graph [Formula: see text] is considered as the center of [Formula: see text] and it is symbolized by [Formula: see text], i.e., [Formula: see text]. A graph may have one or more central vertices. This paper develops an optimal algorithm to compute the diameter, radius and central node (s) of the permutation graph having [Formula: see text] node points in [Formula: see text] time. We have also established a tight relation between radius and diameter of permutation graphs.