It states that every drawing in the plane of a non-planar graph contains a pair of independent edges (not both sharing an endpoint) that cross each other an odd number of times.
[1] The result is named after Haim Hanani, who proved in 1934 that every drawing of the two minimal non-planar graphs K5 and K3,3 has a pair of edges with an odd number of crossings,[2] and after W. T. Tutte, who stated the full theorem explicitly in 1970.
[3] A parallel development of similar ideas in algebraic topology has been credited to Egbert van Kampen, Arnold S. Shapiro, and Wu Wenjun.
[4][5][6][7][8][9] One consequence of the theorem is that testing whether a graph is planar may be formulated as solving a system of linear equations over the finite field of order two.
These equations may be solved in polynomial time, but the resulting algorithms are less efficient than other known planarity tests.