Kuratowski’s theorem

Kuratowski's theorem is a fundamental result in graph theory, a branch of mathematics concerned with networks of points (vertices) connected by lines (edges). The theorem provides a criterion for determining if a graph is planar, meaning it can be drawn on a flat surface without any edges crossing each other except at the vertices.1

Specifically, Kuratowski's theorem states that a graph is nonplanar if and only if it contains a subgraph that is a subdivision of either the complete graph on five vertices (K₅) or the complete bipartite graph on six vertices (K_3,3). These two specific graphs, K₅ and K_3,3, are the smallest nonplanar graphs and serve as building blocks for all other nonplanar graphs.1