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The Fraysseix–Rosenstiehl planarity criterion can be used directly as part of algorithms for planarity testing, while Kuratowski's and Wagner's theorems have indirect applications: if an algorithm can find a copy of K 5 or K 3,3 within a given graph, it can be sure that the input graph is not planar and return without additional computation.
The graph K 3,3, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it cannot be planar. These theorems provide necessary conditions for planarity that are not sufficient conditions, and therefore can only be used to prove a graph is not planar, not that it is planar. If both theorem 1 and 2 fail, other ...
Minor testing (checking whether an input graph contains an input graph as a minor); the same holds with topological minors; Steiner tree, or Minimum spanning tree for a subset of the vertices of a graph. [2] (The minimum spanning tree for an entire graph is solvable in polynomial time.) Modularity maximization [5]
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 ...
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]
In graph theory, a bipolar orientation or st-orientation of an undirected graph is an assignment of a direction to each edge (an orientation) that causes the graph to become a directed acyclic graph with a single source s and a single sink t, and an st-numbering of the graph is a topological ordering of the resulting directed acyclic graph. [1] [2]
Blossom tree (graph theory) Book (graph theory) ... Left-right planarity test; Friendship graph; Frucht graph; G. ... Statistics; Cookie statement;