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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 computational geometry and geometric graph theory, a planar straight-line graph (or straight-line plane graph, or plane straight-line graph), in short PSLG, is an embedding of a planar graph in the plane such that its edges are mapped into straight-line segments. [1] Fáry's theorem (1948) states that every planar graph has this kind of ...
One direction of the characterisation states that every planar graph has a 2-basis. Such a basis may be found as the collection of boundaries of the bounded faces of a planar embedding of the given graph G. If an edge is a bridge of G, it appears twice on a single face boundary and therefore has a zero coordinate in the corresponding vector ...
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.
Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates. The polar coordinate system is extended to three dimensions in two ways: the cylindrical coordinate system adds a second distance coordinate, and the spherical ...
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. [9] Such a drawing is called a plane graph or planar embedding of the graph.
These are the graphs that do not have the utility graph, (which is not Pfaffian) as a graph minor. By Wagner's theorem, the ,-minor-free graphs are formed by gluing together copies of planar graphs and the complete graph along shared edges. The same gluing structure can be used to obtain a Pfaffian orientation for these graphs.
Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K 5 (top) or K 3,3 (bottom) subgraphs. Wagner published both theorems in 1937, [1] subsequent to the 1930 publication of Kuratowski's theorem, [2] according to which a graph is planar if and only if it does not contain as a subgraph a subdivision of one of the same two forbidden ...