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Area of Science:

  • Graph theory
  • Combinatorial mathematics
  • Game theory

Background:

  • Mutual-visibility is a key graph property with applications in network design.
  • Understanding the strategic formation of mutual-visibility sets is computationally challenging.

Purpose of the Study:

  • To introduce and analyze a game-based model for constructing mutual-visibility sets in graphs.
  • To determine the optimal size of mutual-visibility sets under adversarial play.

Main Methods:

  • A two-player game (Builder vs. Blocker) on graph vertices.
  • Analysis of optimal strategies and resulting set sizes.
  • Derivation of bounds and properties for general graphs.

Main Results:

  • Established bounds on the size of mutual-visibility sets formed by optimal play.
  • Characterized game properties and outcomes.
  • Investigated performance on complete multipartite and Hamming graphs.

Conclusions:

  • The game provides a novel framework for studying mutual-visibility.
  • Optimal play leads to predictable bounds on set size.
  • Specific graph structures influence game outcomes.