Center and periphery of lexicographic product of digraphs

Abstract

Let D = (V,E) be a digraph and u, v ∈ V.  The metric maximum distance is defined by md(u,v) = max {d⃗(u,v), d⃗(v,u)}, where d⃗(u,v) denote the length of a shortest directed u − v path in D. The m-eccentricity of a vertex v in D is defined by mecc(v) = max {md(v,u) : u ∈ V(D)}.  The m-center mC(D) of a strongly connected digraph D consists of all the vertices with the minimum m-eccentricity, and the m-periphery mPer(D) consists of all the vertices with the maximum m-eccentricity in D. The relationship between the m-center and the m-periphery of two digraphs and their lexicographic product is studied in this article.