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 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.

scholarly journals Parameterized Complexity of Directed Spanner Problems

Algorithmica ◽  
2021 ◽  
Author(s):  
Fedor V. Fomin ◽  
Petr A. Golovach ◽  
William Lochet ◽  
Pranabendu Misra ◽  
Saket Saurabh ◽  
...  
Keyword(s):  
Directed Graph ◽  
Parameterized Complexity ◽  
Directed Graphs ◽  
Directed Acyclic Graphs ◽  
Polynomial Kernel ◽  
Distortion Parameter ◽  
Spanning Subgraph ◽  
Acyclic Graphs ◽  
Positive Integers ◽  
Additive Distortion

AbstractWe initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most $$m-k$$ m - k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most $$m-k$$ m - k arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size $$\mathcal {O}(k^4t^5)$$ O ( k 4 t 5 ) and can be solved in randomized $$(4t)^k\cdot n^{\mathcal {O}(1)}$$ ( 4 t ) k · n O ( 1 ) time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in $$k^{2k}\cdot n^{\mathcal {O}(1)}$$ k 2 k · n O ( 1 ) time on directed acyclic graphs, (iii) Directed Additive Spanner is $${{\,\mathrm{\mathsf{W}}\,}}[1]$$ W [ 1 ] -hard when parameterized by k for every fixed $$t\ge 1$$ t ≥ 1 even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is $${{\,\mathrm{\mathsf{FPT}}\,}}$$ FPT when parameterized by t and k.

Giant components in three-parameter random directed graphs

1992 ◽  
Vol 24 (4) ◽  
pp. 845-857 ◽  
Author(s):  
Tomasz Łuczak ◽  
Joel E. Cohen
Keyword(s):  
Phase Transition ◽  
Food Webs ◽  
Directed Graph ◽  
Directed Graphs ◽  
Giant Component ◽  
Critical Surface ◽  
Connected Component ◽  
Strongly Connected Component ◽  
Strongly Connected ◽  
Asymptotic Probability

A three-parameter model of a random directed graph (digraph) is specified by the probability of ‘up arrows' from vertexito vertexjwherei < j, the probability of ‘down arrows' fromitojwherei ≥ j,and the probability of bidirectional arrows betweeniandj.In this model, a phase transition—the abrupt appearance of a giant strongly connected component—takes place as the parameters cross a critical surface. The critical surface is determined explicitly. Before the giant component appears, almost surely all non-trivial components are small cycles. The asymptotic probability that the digraph contains no cycles of length 3 or more is computed explicitly. This model and its analysis are motivated by the theory of food webs in ecology.

Stationary vectors of stochastic matrices subject to combinatorial constraints

2015 ◽  
Vol 28 ◽  
Author(s):  
Jane Breen ◽  
Steve Kirkland
Keyword(s):  
Bipartite Graph ◽  
Directed Graph ◽  
Convex Polytopes ◽  
Probability Vector ◽  
Positive Probability ◽  
Stochastic Matrices ◽  
Spanning Subgraph ◽  
Admissible Vector ◽  
Strongly Connected ◽  
Stationary Vector

Given a strongly connected directed graph D, let S_D denote the set of all stochastic matrices whose directed graph is a spanning subgraph of D. We consider the problem of completely describing the set of stationary vectors of irreducible members of S_D. Results from the area of convex polytopes and an association of each matrix with an undirected bipartite graph are used to derive conditions which must be satisfied by a positive probability vector x in order for it to be admissible as a stationary vector of some matrix in S_D. Given some admissible vector x, the set of matrices in S_D that possess x as a stationary vector is also characterised.

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

Algorithms ◽  
2020 ◽  
Vol 13 (12) ◽  
pp. 321
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 Balatonboglár 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 black and white SAT problems. We show that the Balatonboglár model is between the strong and the weak model, and it generates 3-SAT problems, so the Balatonboglár 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 Balatonboglár 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 Balatonboglár 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: the SAT problem and directed graphs.

Giant components in three-parameter random directed graphs

1992 ◽  
Vol 24 (04) ◽  
pp. 845-857 ◽  
Author(s):  
Tomasz Łuczak ◽  
Joel E. Cohen
Keyword(s):  
Phase Transition ◽  
Food Webs ◽  
Directed Graph ◽  
Directed Graphs ◽  
Giant Component ◽  
Critical Surface ◽  
Connected Component ◽  
Strongly Connected Component ◽  
Strongly Connected ◽  
Asymptotic Probability

A three-parameter model of a random directed graph (digraph) is specified by the probability of ‘up arrows' from vertex i to vertex j where i &lt; j, the probability of ‘down arrows' from i to j where i ≥ j, and the probability of bidirectional arrows between i and j. In this model, a phase transition—the abrupt appearance of a giant strongly connected component—takes place as the parameters cross a critical surface. The critical surface is determined explicitly. Before the giant component appears, almost surely all non-trivial components are small cycles. The asymptotic probability that the digraph contains no cycles of length 3 or more is computed explicitly. This model and its analysis are motivated by the theory of food webs in ecology.

Exponential Consensus in Directed Networks of Multi-Agents with Saturated Protocols

2016 ◽  
Vol 16 (02) ◽  
pp. 1650003
Author(s):  
QINGLING WANG ◽  
YUANDA WANG
Keyword(s):  
Lyapunov Function ◽  
Directed Graph ◽  
Spanning Tree ◽  
Directed Graphs ◽  
Consensus Problem ◽  
Directed Networks ◽  
Strongly Connected ◽  
Theoretical Results ◽  
Multi Agents

This paper addresses the exponential consensus problem of single-integrator agents with saturated protocols on directed graphs. By employing an integral Lyapunov function, the exponential consensus problem of single-integrator agents is solved under the directed graph with strongly connected or a spanning tree. The main contribution is that under the directed graph, some conditions for exponential consensus with saturated protocols are first obtained. Finally, two examples are used to illustrate the effectiveness of the theoretical results.

scholarly journals On Directed Versions of the Hajnal–Szemerédi Theorem

2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Perfect Matchings ◽  
Large Order ◽  
Vertex Disjoint ◽  
Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

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