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About 4,492 papers for “Digraph”

Digraph homomorphisms on \(Z_n\)-digraph

Jimly Manuel, Bindhu K Thomas · Malaya Journal of Matematik · 2020

A graph homomorphism is a mapping between two graphs that respect their structure. In this paper we develop some results related to digraph homomorphisms for the class of \({\overrightarrow{Z_n}}^{-}\)-digraphs. We will begin by giving some standard definitions, then expanding our focus to specifically study different types of digraph homomorphisms. In particular we will be discussing the idea of automorphism, isomorphism, weak homomorphism and its relation with the corresponding group homomorphisms defined on \(Z_n\).

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ISOMORFISMA DIGRAPH EKSENTRIS DENGAN DIGRAPH ASALNYA

IDA RIYAWATI, 0800112306 · Universitas Airlangga Repository (Universitas Airlangga) · 2005

Digraph sikel Cn adalah digraph yang paling sederhana yang digraph
\neksentrisnya isomorfis dengan digraph asalnya. Sedangkan digraph lengkap Kn adalah digraph komplit yang digraph eksentrisnya isomorfis bahkan sama dengan digraph asalnya, sehingga untuk mencari digraph-liigraph lain yang digraph eksentrisnya isomorfis dengan digraph asalnya dapat dieari dari digraph sikel Cn ditambah dengan garis berarah dan digraph lengkap Kn dengan dikurangi garis berarah.

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Digraph Decompositions and Monotonicity in Digraph Searching

Stephan Kreutzer, Sebastian Ordyniak · ArXiv.org · 2008

We consider monotonicity problems for graph searching games. Variants of these games - defined by the type of moves allowed for the players - have been found to be closely connected to graph decompositions and associated width measures such as path- or tree-width. Of particular interest is the question whether these games are monotone, i.e. whether the cops can catch a robber without ever allowing the robber to reach positions that have been cleared before. The monotonicity problem for graph searching games has intensely been studied in the literature, but for two types of games the problem was left unresolved. These are the games on digraphs where the robber is invisible and lazy or visible and fast. In this paper, we solve the problems by giving examples showing that both types of games are non-monotone. Graph searching games on digraphs are closely related to recent proposals for digraph decompositions generalising tree-width to directed graphs. These proposals have partly been motivated by attempts to develop a structure theory for digraphs similar to the graph minor theory developed by Robertson and Seymour for undirected graphs, and partly by the immense number of algorithmic results using tree-width of undirected graphs and the hope that part of this success might be reproducible on digraphs using a directed tree-width. Unfortunately the number of applications for the digraphs measures introduced so far is still small. We therefore explore the limits of the algorithmic applicability of digraph decompositions. In particular, we show that various natural candidates for problems that might benefit from digraphs having small directed tree-width remain NP-complete even on almost acyclic graphs.

Cited by 22Open access

Digraph Decompositions and Monotonocity in Digraph Searching

Stephan Kreutzer, Sebastian Ordyniak · Oxford University Research Archive (ORA) (University of Oxford) · 2008

We consider Cops and Robber games on digraphs and their corresponding decompositions.In particular we show non-monotonicity of two important variants of these games, namely the variants underlying DAG-and Kelly-decompositions. Furthermore we explore the limits of the algorithmic applicability of digraph decompositions and show that various natural candidates for problems that might benefit from digraphs having small "directed width" remain NP-complete even on almost acyclic graphs.Despite the considerable interest and the large number of results in this field, two cases have so far resisted any attempts to solve the monotonicity problem -the Cops and Robber game with a visible, fast robber and the game with an invisible, lazy robber, both played on digraphs.In this paper,we solve the problems by showing that both games are non-monotone. Digraph decompositions.In recent years, attempts have been made to generalise the notion of treedecompositions and their algorithmic applications to directed graphs.Clearly, we can define the treewidth of a directed graph as the tree-width of the undirected graph we get by forgetting the direction of edges, a process which leads to some loss of information.This loss may be significant, if the algorithmic problems we are interested in are inherently directed.A good example is the problem of detecting Hamiltonian cycles.While we know that this can be solved easily on graphs with small tree-width, there are directed graphs with very simple connectivity structure which have large tree-width.Therefore, several proposals have been made to extend the notions of tree-decompositions and tree-width to directed graphs (see [24,17,4,6,25,16]).In particular, Reed [24] and Johnson, Robertson, Seymour, and Thomas [17] introduce the notion of directed tree-width and they show that Hamiltonicity can be solved for graphs of bounded directed tree-width in polynomial time.Following this initial paper, several alternative definitions of directed graph decompositions have been proposed, with the aim of overcoming some shortcomings of the original definition.Berwanger, Dawar, Hunter, Kreutzer [5], and Obdrlek [23] introduce the notion of DAG-width and Hunter and Kreutzer [16] introduce the notion of Kelly-width.All three proposals are supported by algorithmic applications and various equivalent characterisations in terms of obstructions, elimination orderings, and, in particular, variants of Cops and Robber games on directed graphs.However, so far the algorithmic applications are restricted to few classes of problems, in particular the problem of finding disjoint paths, Hamiltonian-cycles and similar linkage problems, and certain problems in relation to combinatorial games (parity games) played on graphs that are motivated by the theory of computer-aided verification.Whereas the tree-width of undirected graphs has been employed to solve a huge number of problems on graphs of small tree-width, the algorithmic theory of directed graph decompositions is not nearly as rich.It is an obvious question whether this is due to the fact that digraph decompositions are a relatively new field of research, where the fundamental machinery first needs to be developed, or whether this is due to a general limitation of this approach to algorithms on digraphs.In this paper we systematically explore the range of algorithmic applicability of digraph decompositions.For this, we look at typical NP-complete problems on graphs -as they can be found in [15] for instance -and identify those that are "suitable"for this approach, where by "suitable" we mean that the problems should be NP-complete in general but be tractable on acyclic graphs.The reason for the latter is that all digraph decompositions proposed so far measure in some way the similarity of a graph to being acyclic.In particular, acyclic graphs have small width in all of these measures.Hence, if a problem is already hard on acyclic digraphs, there is no point in studying the effect of digraph decompositions on this problem.We then identify representatives for the various types of "suitable" problems and ask whether they can be solved in polynomial time on graphs of small directed tree-width, Kelly-or DAG-width, or directed path-width.The results we present in Section 4 show that the border for algorithmic applicability of digraph decompositions is rather tight.Essentially, as far as classical graph theoretical problems are concerned, disjoint paths and Hamiltonian-cycles can be detected efficiently on graphs of small directed tree-width, but all other problems we considered such as Minimum Equivalent Subgraph, Feedback Vertex Set (FVS), Feedback Arc Set, Graph Grundy Numbering, and several others are NP-complete even on graphs with a very low global connectivity and thus very low directed path or tree-width.Organisation.The paper is organised as follows.In Section 2 we briefly recall basic notions from graph and game theory needed in the sequel.In Section 3 we give a formal description of Cops and Robber games and present the first main result of this paper, the non-monotonicity of the two types of games mentioned above.In Section 4 we explore the algorithmic boundaries of the digraph decomposi-

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Matrix Digraph

Ghosh, Sayani, Meyer, Bradley Stewart · Zenodo (CERN European Organization for Nuclear Research) · 2025

matrix_digraph is a repository that constains codes and data for computing determinants via matrix-tree theorem methods. Updates: v1.2.0 adds partitioned rooting for calculation of the factored determinant.

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Digraph applications

Haupt, Brian Thomas. · K-State Research Exchange (Kansas State University) · 2015

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Matrix Digraph

Ghosh, Sayani, Meyer, Bradley Stewart · Zenodo (CERN European Organization for Nuclear Research) · 2025

matrix_digraph is a repository that constains codes and data for computing determinants via matrix-tree theorem methods. Updates: v1.2.1 fixes the citation file.

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ECCENTRIC DIGRAPH

CHU YINAN · National University of Singapore · 2011

Bachelor's

Cited by 0

Digraph colouring

Lucas Picasarri-Arrieta · theses.fr (ABES) · 2024

Cette thèse est dédiée à l'étude de la dicoloration, une notion de coloration pour les digraphes introduite par ErdH{o}s et Neumann-Lara à la fin des années 1970, ainsi que le paramètre qui lui est associé, à savoir le nombre dichromatique. Au cours des dernières décennies, ces deux notions ont permis de généraliser de nombreux résultats classiques de coloration de graphes. Nous commençons par donner différentes bornes sur le nombre dichromatique des digraphes dont le graphe sous-jacent est un graphe cordal. Ensuite, nous améliorons la borne donnée par le théorème de Brooks pour les digraphes sans arcs antiparallèles et introduisons une notion de dégénérescence variables pour les digraphes, ce qui nous permet de prouver une version plus générale du théorème de Brooks.Nous étudions ensuite les digraphes k-dicritiques, c'est-à-dire les obstructions minimales à la (k-1)-dicolorabilité. En particulier, nous généralisons un résultat de Gallai au cas dirigé, et nous prouvons une conjecture de Kostochka et Stiebitz dans le cas particulier k=4. Nous discutons également la densité maximum de tels digraphes, et prouvons qu'il n'y a qu'un nombre fini de digraphes semi-complets 3-dicritiques. On donne par la suite certains résultats structurels sur les digraphes dicritiques de grand ordre.Enfin, nous étudions la notion de redicoloration pour les digraphes. En particulier, nous prouvons que de nombreux résultats soutenant la conjecture de Cereceda se généralisent au cas dirigé.

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Matrix Digraph

Ghosh, Sayani, Meyer, Bradley Stewart · Zenodo (CERN European Organization for Nuclear Research) · 2025

matrix_digraph is a repository that constains codes and data for computing determinants via matrix-tree theorem methods. Updates: v1.2.1 fixes the citation file.

Cited by 0Open access

Digraph redicolouring

Nicolás Bousquet, Frédéric Havet, Nicolas Nisse et al. · European Journal of Combinatorics · 2023

Cited by 3Open access

Line digraph iterations and the (d,k) digraph problem

Fiol Mora, Miquel Àngel, Andrés Yebra, José Luis, Alegre de Miguel, Ignacio · RECERCAT (Consorci de Serveis Universitaris de Catalunya)

This paper studies the behavior of the diameter and\nthe average distance between vertices of the line digraph of a given\ndigraph. The results obtained are then applied to the so-called\n(d, k) digraph problem, that is, to maximize the number of vertices\nin a digraph of maximum out-degree d and diameter k. By line\ndigraph iterations it is possible to construct digraphs with a number\nof vertices larger than (d2 - l)/d2 times the (nonattainable)\nMoore bound. In particular, this solves the (d, k) digraph problem\nfor k = 2. Also, the line digraph technique provides us with a\nsimple local routing algorithm for the corresponding networks.

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