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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 ...
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 ]
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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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;
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,
In category theory, a branch of mathematics, Mac Lane's coherence theorem states, in the words of Saunders Mac Lane, “every diagram commutes”. [1] But regarding a result about certain commutative diagrams, Kelly is states as follows: "no longer be seen as constituting the essence of a coherence theorem". [ 2 ]
Mac Lane cofounded category theory with Samuel Eilenberg, which enables a unified treatment of mathematical structures and of the relations among them, at the cost of breaking away from their cognitive grounding. Nevertheless, his views—however informal—are a valuable contribution to the philosophy and anthropology of mathematics. [2]