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A complete bipartite graph of K 4,7 showing that Turán's brick factory problem with 4 storage sites (yellow spots) and 7 kilns (blue spots) requires 18 crossings (red dots) For any k, K 1,k is called a star. [2] All complete bipartite graphs which are trees are stars. The graph K 1,3 is called a claw, and is used to define the claw-free graphs ...
However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph K 5 plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither K 5 nor the complete bipartite graph K 3,3 as a subdivision, and by ...
In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Outerplanar graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two forbidden minors K 4 and K 2,3, or by their Colin de Verdière graph invariants. They have Hamiltonian ...
Every maximal planar graph on more than 3 vertices is at least 3-connected. [6] If a maximal planar graph has v vertices with v > 2, then it has precisely 3v – 6 edges and 2v – 4 faces. Apollonian networks are the maximal planar graphs formed by repeatedly splitting triangular faces into triples of smaller triangles.
The complete graphs on three and four vertices, K 3 and K 4, are both Apollonian networks. K 3 is formed by starting with a triangle and not performing any subdivisions, while K 4 is formed by making a single subdivision before stopping. The Goldner–Harary graph is an Apollonian network that forms the smallest non-Hamiltonian maximal planar ...
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. If is a graph that contains a subgraph that is a subdivision of or ,, then is known as a Kuratowski subgraph of . [1]
That is, if there exists a collection of k planar graphs, all having the same set of vertices, such that the union of these planar graphs is G, then the thickness of G is at most k. [1] [2] In other words, the thickness of a graph is the minimum number of planar subgraphs whose union equals to graph G. [3] Thus, a planar graph has thickness one.
Baker's technique covers a planar graph with a constant number of -outerplanar graphs and uses their low treewidth in order to quickly approximate several hard graph optimization problems. [ 2 ] In connection with the GNRS conjecture on metric embedding of minor-closed graph families, the k {\displaystyle k} -outerplanar graphs are one of the ...
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