Left-right planarity test

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.

[6] For any depth-first search of a graph G, the edges encountered when discovering a vertex for the first time define a depth-first search tree T of G. This is a Trémaux tree, meaning that the remaining edges (the cotree) each connect a pair of vertices that are related to each other as an ancestor and descendant in T. Three types of patterns can be used to define two relations between pairs of cotree edges, named the T-alike and T-opposite relations.

In the next two figures, the edges with the same labels are T-opposite, meaning that they will be on different sides of the tree in every planar drawing.

This characterization immediately leads to an (inefficient) planarity test: determine for all pairs of edges whether they are T-alike or T-opposite, form an auxiliary graph that has a vertex for each connected component of T-alike edges and an edge for each pair of T-opposite edges, and check whether this auxiliary graph is bipartite.

Making this algorithm efficient involves finding a subset of the T-alike and T-opposite pairs that is sufficient to carry out this method without determining the relation between all edge pairs in the input graph.