Minimal strong digraphs

We introduce adequate concepts of expansion of a digraph to obtain a sequential construction of minimal strong digraphs. We obtain a characterization of the class of minimal strong digraphs whose expansion preserves the property of minimality. We prove that every minimal strong digraph of order nmayor que=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

Structural and spectral properties of minimal strong digraphs

EL artículo se centra en las propiedades estructurales y espectrales de los digrafos fuertemente conexos minimales, mediante la comparación de sus propiedades con las propiedades de los árboles. Este análisis incluye dos propiedades nuevas la primera da cotas para los coeficientes de los polinomios característicos de los árboles, y conjetura que esas cotas se generalizan para digrafos fuertemente conexos minimales. Como caso particular, probamos que el término independiente de tale polinomios debe ser -1, 0 o 1. La segunda establece que todo digrafo fuertemente conexo minimal puede descomponerse en un arbol generador dirigido con raíz, y un bosque de árboles con raíz inversos. En nuestra opinión, las analogías descritas entre árboles y digrafos fuertemente conexos minimales suponen un cambio significativo sobre el punto de vista acerca de estos últimos. Abstract In this article, we focus on structural and spectral properties of minimal strong digraphs (MSDs). We carry out a comparative study of properties of MSDs versus trees. This analysis includes two new properties. The first one gives bounds on the coefficients of characteristic polynomials of trees (double directed trees), and conjectures the generalization of these bounds to MSDs. As a particular case, we prove that the independent coemcient of the characteristic polynomial of a tree or an MSD must be — 1, 0 or 1. For trees, this fact means that a tree has at most one perfect matching; for MSDs, it means that an MSD has at most one covering by disjoint cycles. The property states that every MSD can be decomposed in a rooted spanning tree and a forest of reversed rooted trees, as factors. In our opinión, the analogies described suppose a significative change in the traditional point of view about this class of digraphs

Structural properties of minimal strong digraphs versus trees

Producción CientíficaIn 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).Ministerio de Economía, Industria y Competitividad ( grant MTM2015-65764-C3-1-P

Bounds of the longest directed cycle length for minimal strong digraphs

AbstractIn this paper we present the upper and lower bounds of the longest directed cycle length for minimal strong digraphs in terms of the numbers of vertices and arcs. These bounds are both sharp. In addition, we give analogous results for minimal 2-edge connected graphs

Hitting minors, subdivisions, and immersions in tournaments

The Erd\H{o}s-P\'osa property relates parameters of covering and packing of combinatorial structures and has been mostly studied in the setting of undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim, and Seymour to show that, for every directed graph $H$ (resp. strongly-connected directed graph $H$), the class of directed graphs that contain $H$ as a strong minor (resp. butterfly minor, topological minor) has the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove that if $H$ is a strongly-connected directed graph, the class of directed graphs containing $H$ as an immersion has the edge-Erd\H{o}s-P\'osa property in the class of tournaments.Comment: Accepted to Discrete Mathematics & Theoretical Computer Science. Difference with the previous version: use of the DMTCS article class. For a version with hyperlinks see the previous versio