Antipodal graphs and digraphs
To a given graph \(G\) we can define the antipodal graph \(A(G)\) which has the same vertex set as \(G\) and an edge \((u,v)\) is in \(A(G)\) if \(d(u,v)\) is equal to the diameter of \(G\). (If \(G\) is disconnected, then diam \(G=\infty.)\) This definition is extended to a digraph too. It is shown that a digraph \(D\) is an antipodal digraph if and only if \(D\) is the antipodal digraph of its complement. This result generalizes a known characterization of antipodal graphs. Authors also study self-antipodal digraphs (i.e. \(A(D)\) is isomorphic to \(D)\) and self-antipodal graphs. They give some interesting results. Note that in the literature other graphs were also called antipodal graphs, see e.g. \textit{A. Berman} and \textit{A. Kotzig} [Discrete Math. 69, No. 2, 107-114 (1988; Zbl 0667.05048)], \textit{A. Berman}, \textit{A. Kotzig} and \textit{G. Sabidussi} [Contemporary methods in graph theory. In honour of Prof. Dr. K. Wagner, 137-150 (1990; Zbl 0722.05042)] and \textit{K. Fukuda} and \textit{K. Handa} [Discrete Math. 111, No. 1-3, 245-256 (1993; Zbl 0782.05067)].
scientific article; zbMATH DE number 436741