Structural properties of minimal strong digraphs versus trees

Abstract

In this article, we focus on structural properties of minimal strong digraphs (MSDs). We carry out a comparative study of properties of MSDs versus (undirected) trees. For some of these properties, we give the matrix version, regarding nearly reducible matrices. We give bounds for the coefficients of the characteristic polynomial corresponding to the adjacency matrix of trees, and we conjecture bounds for MSDs. We also propose two different representations of an MSD in terms of trees (the union of a spanning tree and a directed forest; and a double directed tree whose vertices are given by the contraction of connected Hasse diagrams).

Structure of cycles in minimal strong digraphs

“…In this paper we use some concepts and basic results about graphs that are described below, in order to fix the notation [2,5,6].…”

“…A vértex nona digraph is called a cut point if the deletion of v and all its incident ares yields a disconnected digraph. Some basic properties concerning MDSs can be found in [2,3,5,6]. We summarize some of them: In an MSD with n vértices and m ares, n < m < 2(n -1); the contraction of a cycle in an MSD preserves the minimality, that is, it produces another MSD; henee, if we contract a strongly connected subdigraph in a minimal digraph, the resulting digraph is also minimal; each MSD of order n > 2 has at least two linear vértices; in an MSD with exactly two linear vértices, each of them belongs to a unique cycle; furthermore, these eyeles are C 2 or C 3 .The next result will be explicitly used insome of our proofs.…”

“…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.…”

“…The configuration of cycles of an MSD determines, among many other things, the characteristic polynomial of its adjaceney matrix and, therefore, it is very important for the spectral theory of this family of graphs [5,6]. It is also related to algorithmic problems that, restricted to MSDs, can present unexpected properties, given the strong analogy that exists between this family of graphs and trees [6]. In this way it is interesting to study the length / of the longest cycle in an MSD, relate this parameter with the number of ares and design an algorithm that calculates it quickly.…”

Structural and spectral properties of minimal strong digraphs

“…We carry out a comparative study of properties of MSDs versus trees. An extended versión of this work can be found in [7]. A double directed tree is the digraph obtained from a tree by replacing each edge {u,v} with the two ares (u,v) and (v,u).…”