On the Existence and Determination of Satisfactory Partitions in a Graph 1st edition by Cristina Bazgan, Zsolt Tuz, Daniel Vanderpooten ISBN 3540206958 9783540206958

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ISBN 10:       3540206958
ISBN 13: 978-3540206958
Author:         Cristina Bazgan, Zsolt Tuz, Daniel Vanderpooten 
 The Satisfactory Partition problem consists in deciding if a given graph has a partition of its vertex set into two nonempty sets V ₁,V ₂ such that for each vertex v, if v ∈ V _i then <span id="MathJax-Element-1-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; font-size-adjust: none; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="dVi(v)≥s(v)”>dVi(v)≥s(v), where s(v)≤ d(v) is a given integer-valued function. This problem was introduced by Gerber and Kobler [EJOR 125 (2000), 283–291] for <span id="MathJax-Element-2-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; font-size-adjust: none; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="s=⌈d2⌉”>s=⌈d2⌉. In this paper we study the complexity of this problem for different values of s.

On the Existence and Determination of Satisfactory Partitions in a Graph 1st Table of contents:

• Introduction

  • 1.1 Overview of Graph Partitioning Problems
  • 1.2 Motivating the Study of Satisfactory Partitions
  • 1.3 Key Concepts in Graph Theory
  • 1.4 Importance and Applications of Graph Partitioning
  • 1.5 Structure of the Paper

• Graph Theory Preliminaries

  • 2.1 Basic Definitions in Graph Theory
  • 2.2 Types of Graphs (e.g., Directed, Undirected, Bipartite)
  • 2.3 Graph Partitioning: Definitions and Challenges
  • 2.4 Common Partitioning Criteria (e.g., Balanced, Cut-Based)
  • 2.5 Previous Work on Graph Partitions and Satisfactory Partitioning

• The Problem of Satisfactory Partitions

  • 3.1 Formal Definition of Satisfactory Partitions
  • 3.2 Criteria for a Satisfactory Partition
  • 3.3 Relationship to Other Graph Partitioning Problems
  • 3.4 Theoretical Characterization of Satisfactory Partitions
  • 3.5 Difficulties and Computational Complexity of Finding Satisfactory Partitions

• Existence of Satisfactory Partitions

  • 4.1 Necessary Conditions for the Existence of a Satisfactory Partition
  • 4.2 Sufficient Conditions for Satisfactory Partitions
  • 4.3 Theorems on Existence and Non-Existence
  • 4.4 Graph Classes with Guaranteed Satisfactory Partitions
  • 4.5 Counterexamples: Graphs without Satisfactory Partitions

• Algorithms for Determining Satisfactory Partitions

  • 5.1 Overview of Graph Partitioning Algorithms
  • 5.2 Exact Algorithms for Satisfactory Partitions
  • 5.3 Approximation Algorithms for Finding Satisfactory Partitions
  • 5.4 Heuristic Methods and Practical Approaches
  • 5.5 Algorithmic Complexity and Performance Analysis

• Special Cases and Applications

  • 6.1 Satisfactory Partitions in Planar Graphs
  • 6.2 Bipartite Graphs and Satisfactory Partitions
  • 6.3 Partitioning with Respect to Specific Constraints (e.g., Edge Cuts, Vertex Properties)
  • 6.4 Applications in Network Design and Social Networks
  • 6.5 Applications in Computational Biology (e.g., Clustering, Genome Mapping)

• Graph Partitioning in Optimization

  • 7.1 Optimization Models for Graph Partitioning
  • 7.2 Min-Cut and Max-Cut Problems in the Context of Satisfactory Partitions
  • 7.3 Multi-Objective Partitioning: Balancing Multiple Criteria
  • 7.4 Approximation Boundaries and Techniques
  • 7.5 Partitioning with Edge and Vertex Weightings

• Advanced Topics in Graph Partitioning

  • 8.1 Partitioning Dynamic Graphs (e.g., Graphs with Changing Topology)
  • 8.2 Distributed and Parallel Algorithms for Graph Partitioning
  • 8.3 Randomized Algorithms and Probabilistic Techniques
  • 8.4 Online Partitioning Problems

• Open Problems and Future Directions

  • 9.1 Unsolved Problems in Graph Partitioning Theory
  • 9.2 Exploring New Partitioning Criteria
  • 9.3 Extensions to Hypergraphs and Multigraphs
  • 9.4 Potential Applications in New Fields (e.g., Machine Learning, AI)
  • 9.5 Directions for Future Research

• Conclusion

  • 10.1 Summary of Key Findings
  • 10.2 Contributions to Graph Theory and Partitioning Algorithms
  • 10.3 Final Thoughts and Implications for Practice

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