Wagner's Theorem

Wagner's Theorem provides a way to identify when a graph (a network of points connected by lines) cannot be drawn without crossing lines, specifically focusing on the complete graph with five points (K₅) and the complete bipartite graph with six points divided evenly (K₃,₃). It states that a graph is "non-planar"—meaning it can't be laid out on a flat surface without crossings—if and only if it contains a smaller part that looks like one of these two complex graphs. Essentially, it's a criterion for recognizing inherently tangled networks.