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A planar graph is said to be convex if all of its faces (including the outer face) are convex polygons. Not all planar graphs have a convex embedding (e.g. the complete bipartite graph K 2,4). A sufficient condition that a graph can be drawn convexly is that it is a subdivision of a 3-vertex-connected planar graph.
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 ...
The Goldner–Harary graph is a planar graph: it can be drawn in the plane with none of its edges crossing. When drawn on a plane, all its faces are triangular, making it a maximal planar graph. As with every maximal planar graph, it is also 3-vertex-connected: the removal of any two of its vertices leaves a connected subgraph.
Since such graphs have a unique embedding (up to flipping and the choice of the external face), the next bigger graph, if still planar, must be a refinement of the former graph. This allows to reduce the planarity test to just testing for each step whether the next added edge has both ends in the external face of the current embedding.
In the mathematical field of graph theory, Fáry's theorem states that any simple, planar graph can be drawn without crossings so that its edges are straight line segments. That is, the ability to draw graph edges as curves instead of as straight line segments does not allow a larger class of graphs to be drawn.
A clique-sum of two planar graphs and the Wagner graph, forming a K 5-free graph. In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite graph is planar if and only if its minors include neither K 5 (the complete graph on five vertices) nor K 3,3 ...
In the mathematical field of graph theory, planarization is a method of extending graph drawing methods from planar graphs to graphs that are not planar, by embedding the non-planar graphs within a larger planar graph. [1] [2]
In mathematics and computer science, graph theory is the study of graphs, ... For a planar graph, the crossing number is zero by definition. Drawings on surfaces ...