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The classic path addition method of Hopcroft and Tarjan [1] was the first published linear-time planarity testing algorithm in 1974. An implementation of Hopcroft and Tarjan's algorithm is provided in the Library of Efficient Data types and Algorithms by Mehlhorn, Mutzel and Näher.
In graph theory, a branch of mathematics, the left-right planarity test or de Fraysseix–Rosenstiehl planarity criterion [1] is a characterization of planar graphs based on the properties of the depth-first search trees, published by de Fraysseix and Rosenstiehl (1982, 1985) [2] [3] and used by them with Patrice Ossona de Mendez to develop a linear time planarity testing algorithm.
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 ...
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. [1] [2] Such a drawing is called a plane graph, or a planar embedding of the graph.
The nature of a witness value often depends on the type of mathematical calculation being performed. For LEDA's planarity testing function, If the graph is planar, a combinatorial embedding is produced as a witness. If not, a Kuratowski subgraph is returned. These values can then be passed directly to checker functions to confirm their validity.
A Kuratowski subgraph of a nonplanar graph can be found in linear time, as measured by the size of the input graph. [2] This allows the correctness of a planarity testing algorithm to be verified for nonplanar inputs, as it is straightforward to test whether a given subgraph is or is not a Kuratowski subgraph. [3]
The Hopcroft–Tarjan planarity testing algorithm was the first linear-time algorithm for planarity testing. [11] Tarjan has also developed important data structures such as the Fibonacci heap (a heap data structure consisting of a forest of trees), and the splay tree (a self-adjusting binary search tree; co-invented by Tarjan and Daniel Sleator).
Mac Lane's planarity criterion uses this idea to characterize the planar graphs in terms of the cycle bases: a finite undirected graph is planar if and only if it has a sparse cycle basis or 2-basis, [3] a basis in which each edge of the graph participates in at most two basis cycles. In a planar graph, the cycle basis formed by the set of ...