Equivalence of strongly connected graphs and black-and-white 2-SAT problems

Bíró, Csaba; Kusper, Gábor

doi:10.18514/mmn.2018.2140

Cited by 9 publications

(9 citation statements)

References 16 publications

(22 reference statements)

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“…In our previous model [1], which we call in this paper the strong model of communication graphs, edges are represented by logical implication: Let us assume that the graph D is ({a,b,c},{(a, b), (a, c)}); i.e., D is a graph with three vertices, a, b and c, and with two edges, from a to b and from a to c. Thus, the strong model of D is (a =⇒ b) ∧ (a =⇒ c).…”

Section: The Strong Model Of Communication Graphsmentioning

confidence: 99%

“…Furthermore, SM is a black and white SAT problem iff D is strongly connected; see Theorem 1 in [1]. Since we use this theorem several times herein, we recall it: Theorem 1 (Theorem 1 from [1]). Let D be a communication graph.…”

Section: The Strong Model Of Communication Graphsmentioning

confidence: 99%

“…In a previous paper [1] we showed how to convert a directed graph into a 2-SAT problem. In this paper, we aim to translate directed graphs into 3-SAT problems.…”

Section: Introductionmentioning

confidence: 99%

“…In logic the most natural representation of an edge of a directed graph, say a → b, is to use implication-i.e., a =⇒ b; i.e., the edge a → b can be represented by the binary clause: (¬a ∨ b). We used that representation in a previous paper [1] and we were able to prove that the SAT representation of a strongly connected directed graph is a black and white 2-SAT problem. It is not a surprise that the SAT representation is a 2-SAT problem, since each edge can be represented by a binary clause.…”

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confidence: 99%

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

Kusper

Bíró

2020

Algorithms

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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.

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Section: The Strong Model Of Communication Graphsmentioning

confidence: 99%

Section: The Strong Model Of Communication Graphsmentioning

confidence: 99%

“…In a previous paper [1] we showed how to convert a directed graph into a 2-SAT problem. In this paper, we aim to translate directed graphs into 3-SAT problems.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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

Kusper

Bíró

2020

Algorithms

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“…In this paper we assume that the communication graph is strongly connected, the cost of communication between each node is constant (we do not use weights), and the network consists of homogeneous nodes. To test the new metrics we used our own representation [7] and SAT solver [10].…”

Section: Preliminariesmentioning

confidence: 99%

Somek-hop based graph metrics and noderanking in wireless sensor networks

Bíró¹,

Kusper²

2019

AMI

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Node localization and ranking is an essential issue in wireless sensor networks (WSNs). We model WSNs by communication graphs. In our interpretation a communication graph can be directed, in case of heterogeneous sensor nodes, or undirected, in case of homogeneous sensor nodes, and must be strongly connected. There are many metrics to characterize networks, most of them are either global ones or local ones. The local ones consider only the immediate neighbors of the observed nodes. We are not aware of a metric which considers a subgraph, i.e., which is between global and local ones. So our main goal was to construct metrics that interpret the local properties of the nodes in a wider environment. For example, how dense the environment of the given node, or in which extent it can be relieved within its environment. In this article we introduce several novel-hop based density and redundancy metrics: Weighted Communication Graph Density (), Relative Communication Graph Density (ℛ), Weighted Relative Communication Graph Density (ℛ), Communication Graph Redundancy (ℛ), Weighted Communication Graph Redundancy (ℛ). We compare them to known graph metrics, and show that they can be used for node ranking.

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Crazy Truth-Teller–Liar Puzzles

Alzboon

Nagy

2021

Axiomathes

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Equivalence of strongly connected graphs and black-and-white 2-SAT problems

Cited by 9 publications

References 16 publications

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

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

Somek-hop based graph metrics and noderanking in wireless sensor networks

Crazy Truth-Teller–Liar Puzzles

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