scholarly journals Multi-Eulerian Tours of Directed Graphs

10.37236/5588 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Matthew Farrell ◽  
Lionel Levine
Keyword(s):  
Connected Graph ◽  
Directed Graphs ◽  
Minimal Length ◽  
Closed Path ◽  
Directed Edge ◽  
Simple Generalization ◽  
Strongly Connected ◽  
Eulerian Tours ◽  
Eulerian Tour

Not every graph has an Eulerian tour. But every finite, strongly connected graph has a multi-Eulerian tour, which we define as a closed path that uses each directed edge at least once, and uses edges e and f the same number of times whenever tail(e)=tail(f). This definition leads to a simple generalization of the BEST Theorem. We then show that the minimal length of a multi-Eulerian tour is bounded in terms of the Pham index, a measure of 'Eulerianness'.

scholarly journals Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

2020 ◽  
Author(s):  
Gábor Kusper ◽  
Csaba Biró
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Deep Insight ◽  
Communication Graph ◽  
Sat Problem ◽  
Weak Model ◽  
Know How ◽  
Black And White ◽  
Communication Graphs ◽  
Strongly Connected

In a previous paper we defined the Black-and-White SAT problem which has exactly two solutions, where each variable is either true or false. We showed that Black-and-White $2$-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonbogl\'{a}r model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a Black-and-White SAT problem. We prove a powerful theorem, the so called Transitions Theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as Blask-and-White SAT problems. We show that the Balatonbogl\'{a}r model is between the strong and the weak model, and it generates $3$-SAT problems, so the Balatonbogl\'{a}r model represents strongly connected communication graphs as Black-and-White $3$-SAT problems. Our motivation to study these models is the following: The strong model generates a $2$-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonbogl\'{a}r model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonbogl\'{a}r model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: SAT problem and directed graphs.

Strongly Connected Spanning Subgraph for Almost Symmetric Networks

2017 ◽  
Vol 27 (03) ◽  
pp. 207-219
Author(s):  
A. Karim Abu-Affash ◽  
Paz Carmi ◽  
Anat Parush Tzur
Keyword(s):  
Directed Graph ◽  
Euclidean Distance ◽  
Directed Graphs ◽  
Minimum Weight ◽  
Shortest Paths ◽  
Strong Connectivity ◽  
Spanning Subgraph ◽  
Strongly Connected ◽  
Set Of Points ◽  
Symmetric Networks

In the strongly connected spanning subgraph ([Formula: see text]) problem, the goal is to find a minimum weight spanning subgraph of a strongly connected directed graph that maintains the strong connectivity. In this paper, we consider the [Formula: see text] problem for two families of geometric directed graphs; [Formula: see text]-spanners and symmetric disk graphs. Given a constant [Formula: see text], a directed graph [Formula: see text] is a [Formula: see text]-spanner of a set of points [Formula: see text] if, for every two points [Formula: see text] and [Formula: see text] in [Formula: see text], there exists a directed path from [Formula: see text] to [Formula: see text] in [Formula: see text] of length at most [Formula: see text], where [Formula: see text] is the Euclidean distance between [Formula: see text] and [Formula: see text]. Given a set [Formula: see text] of points in the plane such that each point [Formula: see text] has a radius [Formula: see text], the symmetric disk graph of [Formula: see text] is a directed graph [Formula: see text], such that [Formula: see text]. Thus, if there exists a directed edge [Formula: see text], then [Formula: see text] exists as well. We present [Formula: see text] and [Formula: see text] approximation algorithms for the [Formula: see text] problem for [Formula: see text]-spanners and for symmetric disk graphs, respectively. Actually, our approach achieves a [Formula: see text]-approximation algorithm for all directed graphs satisfying the property that, for every two nodes [Formula: see text] and [Formula: see text], the ratio between the shortest paths, from [Formula: see text] to [Formula: see text] and from [Formula: see text] to [Formula: see text] in the graph, is at most [Formula: see text].

scholarly journals MASALAH EIGEN DAN EIGENMODE MATRIKS ATAS ALJABAR MIN-PLUS

2021 ◽  
Vol 15 (4) ◽  
pp. 659-666
Author(s):  
Eka Widia Rahayu ◽  
Siswanto Siswanto ◽  
Santoso Budi Wiyono
Keyword(s):  
Connected Graph ◽  
Irreducible Matrix ◽  
Eigenvalues And Eigenvectors ◽  
Communication Graph ◽  
Strongly Connected ◽  
Square Matrix

Eigen problems and eigenmode are important components related to square matrices. In max-plus algebra, a square matrix can be represented in the form of a graph called a communication graph. The communication graph can be strongly connected graph and a not strongly connected graph. The representation matrix of a strongly connected graph is called an irreducible matrix, while the representation matrix of a graph that is not strongly connected is called a reduced matrix. The purpose of this research is set the steps to determine the eigenvalues and eigenvectors of the irreducible matrix over min-plus algebra and also eigenmode of the regular reduced matrix over min-plus algebra. Min-plus algebra has an ispmorphic structure with max-plus algebra. Therefore, eigen problems and eigenmode matrices over min-plus algebra can be determined based on the theory of eigenvalues, eigenvectors and eigenmode matrices over max-plus algebra. The results of this research obtained steps to determine the eigenvalues and eigenvectors of the irreducible matrix over min-plus algebra and eigenmode algorithm of the regular reduced matrix over min-plus algebra

scholarly journals Hamiltonicity in random directed graphs is born resilient

2020 ◽  
Vol 29 (6) ◽  
pp. 900-942 ◽  
Author(s):  
Richard Montgomery
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Directed Edge

AbstractLet $\{D_M\}_{M\geq 0}$ be the n-vertex random directed graph process, where $D_0$ is the empty directed graph on n vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each $$\varepsilon > 0$$ , we show that, almost surely, any directed graph $D_M$ with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most $1/2-\varepsilon$ of both the in- and out-edges incident to each vertex are removed. We say such a directed graph is $(1/2-\varepsilon)$ -resiliently Hamiltonian. Furthermore, for each $\varepsilon > 0$ , we show that, almost surely, each directed graph $D_M$ in the sequence is not $(1/2+\varepsilon)$ -resiliently Hamiltonian.This improves a result of Ferber, Nenadov, Noever, Peter and Škorić who showed, for each $\varepsilon > 0$ , that the binomial random directed graph $D(n,p)$ is almost surely $(1/2-\varepsilon)$ -resiliently Hamiltonian if $p=\omega(\log^8n/n)$ .

scholarly journals Symbolic Coloured SCC Decomposition

2021 ◽  
pp. 64-83
Author(s):  
Nikola Beneš ◽  
Luboš Brim ◽  
Samuel Pastva ◽  
David Šafránek
Keyword(s):  
Systems Biology ◽  
Directed Graphs ◽  
Connected Components ◽  
Worst Case ◽  
Strongly Connected Components ◽  
Binary Decision ◽  
Scientific Disciplines ◽  
Experimental Implementation ◽  
Strongly Connected ◽  
Symbolic Algorithm

AbstractProblems arising in many scientific disciplines are often modelled using edge-coloured directed graphs. These can be enormous in the number of both vertices and colours. Given such a graph, the original problem frequently translates to the detection of the graph’s strongly connected components, which is challenging at this scale.We propose a new, symbolic algorithm that computes all the monochromatic strongly connected components of an edge-coloured graph. In the worst case, the algorithm performs $$O(p\cdot n\cdot \log n)$$ O ( p · n · log n ) symbolic steps, where p is the number of colours and n the number of vertices. We evaluate the algorithm using an experimental implementation based on Binary Decision Diagrams (BDDs) and large (up to $$2^{48}$$ 2 48 ) coloured graphs produced by models appearing in systems biology.

LOCAL GROUPS IN FREE GROUPOIDS SATISFYING CERTAIN MONOID IDENTITIES

2002 ◽  
Vol 12 (01n02) ◽  
pp. 357-369
Author(s):  
ALAIR PEREIRA DO LAGO
Keyword(s):  
Free Group ◽  
Connected Graph ◽  
Generating Set ◽  
Burnside Groups ◽  
Cyclomatic Number ◽  
Strongly Connected

Let G be a (possibly infinite) strongly connected graph and let [Formula: see text] be a set of monoid identities such that any monoid satisfying [Formula: see text] is also a group. Let ℬ be the free groupoid on G satisfying [Formula: see text]. Then, the local groups ℬv, for v ∈ V (G), are all isomorphic to a free group satisfying [Formula: see text]. Furthermore, it is free over a generating set which can be effectively characterized and whose cardinality is the cyclomatic number of the graph G. We also show applications that establish an important connection between free Burnside groups and free Burnside semigroups.

Routing Flow Through a Strongly Connected Graph

Algorithmica ◽  
2002 ◽  
Vol 32 (3) ◽  
pp. 467-473 ◽  
Author(s):  
Erlebach ◽  
Hagerup
Keyword(s):  
Connected Graph ◽  
Strongly Connected ◽  
Flow Through

$C^*$-algebras of directed graphs and group actions

1999 ◽  
Vol 19 (6) ◽  
pp. 1503-1519 ◽  
Author(s):  
ALEX KUMJIAN ◽  
DAVID PASK
Keyword(s):  
Directed Graph ◽  
Free Group ◽  
Connected Graph ◽  
Dimensional Space ◽  
Directed Graphs ◽  
Free Action ◽  
Crossed Product ◽  
C Algebra ◽  
C Algebras ◽  
Morita Equivalent

Given a free action of a group $G$ on a directed graph $E$ we show that the crossed product of $C^* (E)$, the universal $C^*$-algebra of $E$, by the induced action is strongly Morita equivalent to $C^* (E/G)$. Since every connected graph $E$ may be expressed as the quotient of a tree $T$ by an action of a free group $G$ we may use our results to show that $C^* (E)$ is strongly Morita equivalent to the crossed product $C_0 ( \partial T ) \times G$, where $\partial T$ is a certain zero-dimensional space canonically associated to the tree.

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