Kuratowski's theorem on planar graphs

Kuratowski's theorem provides a way to identify when a graph can be drawn on a flat surface without any edges crossing. It states that a graph is planar if and only if it does not contain a specific pair of complicated subgraphs, known as \(K_5\) (a complete graph with 5 vertices) or \(K_{3,3}\) (a bipartite graph connecting two groups of 3 vertices each), when these subgraphs are possibly stretched or compressed but not broken. In essence, these two subgraphs serve as the fundamental obstacles to planarity: if neither appears, the entire graph can be drawn without overlaps.