Plánární

Plánární is a Czech adjective meaning planar, relating to a plane. In mathematics and related fields, plánární describes objects that can be drawn on a single plane without crossing edges, or that exist entirely within a plane. The term is commonly used in geometry and graph theory to distinguish two-dimensional, plane-dependent properties from three-dimensional ones.

Etymology and usage: The word derives from Latin planus, meaning flat, and has entered Czech through historical linguistic development. In technical contexts, plánární is often paired with nouns such as graf (graph) or grafika (graphics) to denote planarity or plane-based representations.

In geometry and graph theory: A planar (plánární) object lies in a plane. In graph theory, a graph is plánární if it can be drawn on the plane so that its edges intersect only at their endpoints, a drawing known as a planar embedding. Planarity influences how networks, circuits, and spatial data can be visualized and processed.

Key results for planar graphs: For a connected planar graph with V vertices, E edges, and F faces, Euler’s formula holds: V − E + F = 2. For simple planar graphs with V ≥ 3, the inequality E ≤ 3V − 6 applies; if the graph is triangle-free, E ≤ 2V − 4. Planar graphs may have multiple distinct planar embeddings, reflecting different face structures.

Non-planarity: A graph is non-planární if no planar embedding exists. Kuratowski’s theorem characterizes planarity by forbidden subgraphs: a graph is planar if and only if it does not contain a subdivision of K5 or K3,3.

Applications and related terms: Planarity is central in network design, circuit layout, cartography, and computational geometry. Related concepts include planar graphs, planarity testing, graph embedding, and the Euler characteristic.