The (Delta,D) and (Delta,N) problems in double-step digraphs with unilateral diameter

Abstract

We study the (D;D) and (D;N) problems for double-step digraphs considering the unilateral distance, which is the minimum between the distance in the digraph and the distance in its converse digraph, obtained by changing the directions of all the arcs. The first problem consists of maximizing the number of vertices N of a digraph, given the maximum degree D and the unilateral diameter D , whereas the second one consists of minimizing the unilateral diameter given the maximum degree and the number of vertices. We solve the first problem for every value of the unilateral diameter and the second one for some infinitely many values of the number of vertices. Miller and Sirán [4] wrote a comprehensive survey about (D;D) and (D;N) problems. In particular, for the double-step graphs considering the standard diameter, the first problem was solved by Fiol, Yebra, Alegre and Valero [3], whereas Bermond, Iliades and Peyrat [2], and also Beivide, Herrada, Balcázar and Arruabarrena [1] solved the (D;N) problem. In the case of the double-step digraphs, also with the standard diameter, Morillo, Fiol and Fàbrega [5] solved the (D;D) problem and provided some infinite families of digraphs which solve the (D;N) problem for their corresponding numbers of vertices