Minimal strong digraphs

Abstract

We introduce adequate concepts of expansion of a digraph to obtain a sequential construction of minimal strong digraphs. We characterize the class of minimal strong digraphs whose expansion preserves the property of minimality. We prove that every minimal strong digraph of order n ≥ 2 is the expansion of a minimal strong digraph of order n − 1 and we give sequentially generative procedures for the constructive characterization of the classes of minimal strong digraphs. Finally we describe algorithms to compute unlabeled minimal strong digraphs and their isospectral classes.2010-AMS Classification: 05C20, 05C40, 05C75.

Leader election in synchronous networks

“…Another possible approach to generate all labeled strongly connected digraphs is to use the minimal digraphs investigated by García-López and Marijun [27].…”

Structure of cycles in minimal strong digraphs

“…A minimal strong digraph (MSD) is a strong digraph in which the deletion of any are yields a non strongly connected digraph. In [5] a compilation of properties of MSDs can be found, and in [6] an update of the catalog of properties is given, together with a comparative analysis between MSDs and non directed trees, from which a striking analogy between the two families of graphs emerges.…”

Structural and spectral properties of minimal strong digraphs

“…We now record a number of basic faets about the strong digraphs (see [6] and the references therein). In an SD of order n > 2, the in-degree and outdegree of the vértices are bigger than or equal to 1.…”

“…This class of digraphs has been considered under different points of view. See, for instance, [4,6].…”

“…In [6], a sequentially generative procedure for the constructive characterization of the classes of MSDs is given. In addition, algorithms to compute unlabeled MSDs and their isospectral classes are described.…”