Fáry

Fáry is a mathematical theorem in graph theory, named after the Hungarian mathematician István Fáry, who first proved it in 1948. The theorem states that every simple planar graph can be embedded in the plane such that its edges are represented by straight line segments, with no crossings. This means that any graph that can be drawn on a plane without edge intersections can be realized with straight-line edges, simplifying the visualization and analysis of planar graphs.

The significance of Fáry's theorem lies in its implications for graph drawing and topological graph theory. It establishes that the class of planar graphs is equivalent to the subset that admits a straight-line drawing, making it easier to study their properties and algorithms related to planarity. The theorem also provides a foundation for various algorithms in graph visualization, network design, and computational geometry.

Fáry's proof is constructive, offering a method to transform any planar embedding into a straight-line embedding, often utilizing techniques such as induction and geometric transformations. Over the years, the theorem has been extended and refined, influencing research in graph theory and related fields.

Overall, Fáry's theorem is a fundamental result that bridges combinatorial and geometric aspects of graph theory, contributing to both theoretical insights and practical applications in areas requiring planar graph representations.