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In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane who published it in 1937. It states that a finite undirected graph is planar if and only if the cycle space of the graph (taken modulo 2) has a cycle basis in which each edge of the graph ...
Mac Lane's planarity criterion, named after Saunders Mac Lane, characterizes planar graphs in terms of their cycle spaces and cycle bases. It states that a finite undirected graph is planar if and only if the graph has a cycle basis in which each edge of the graph participates in at most two basis cycles.
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Saunders Mac Lane (August 4, 1909 – April 14, 2005), born Leslie Saunders MacLane, was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education [ edit ]
Other planarity criteria, that characterize planar graphs mathematically but are less central to planarity testing algorithms, include: Whitney's planarity criterion that a graph is planar if and only if its graphic matroid is also cographic, Mac Lane's planarity criterion characterizing planar graphs by the bases of their cycle spaces,
Mac Lane's planarity criterion gives an algebraic characterization of finite planar graphs, via their cycle spaces; The Fraysseix–Rosenstiehl planarity criterion gives a characterization based on the existence of a bipartition of the cotree edges of a depth-first search tree. It is central to the left-right planarity testing algorithm;
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Every planar graph has an algebraic dual, which is in general not unique (any dual defined by a plane embedding will do). The converse is actually true, as settled by Hassler Whitney in Whitney's planarity criterion: [44] A connected graph G is planar if and only if it has an algebraic dual. The same fact can be expressed in the theory of matroids.