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2. Outerplanar graph - Wikipedia

   en.wikipedia.org/wiki/Outerplanar_graph

   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 cycles if and only if they are biconnected, in which case the outer face forms the unique Hamiltonian cycle.

3. Complete bipartite graph - Wikipedia

   en.wikipedia.org/wiki/Complete_bipartite_graph

   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 ...

4. k-outerplanar graph - Wikipedia

   en.wikipedia.org/wiki/K-Outerplanar_graph

   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 ...

5. Apollonian network - Wikipedia

   en.wikipedia.org/wiki/Apollonian_network

   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 ...

6. Grötzsch graph - Wikipedia

   en.wikipedia.org/wiki/Grötzsch_graph

   In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5. It is named after German mathematician Herbert Grötzsch, who used it as an example in connection with his 1959 theorem that planar triangle-free graphs are 3-colorable.

7. Grötzsch's theorem - Wikipedia

   en.wikipedia.org/wiki/Grötzsch's_theorem

   Naserasr showed that every triangle-free planar graph also has a homomorphism to the Clebsch graph, a 4-chromatic graph. By combining these two results, it may be shown that every triangle-free planar graph has a homomorphism to a triangle-free 3-colorable graph, the tensor product of with the Clebsch

8. Wagner's theorem - Wikipedia

   en.wikipedia.org/wiki/Wagner's_theorem

   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 ...

9. Robertson–Seymour theorem - Wikipedia

   en.wikipedia.org/wiki/Robertson–Seymour_theorem

   Wagner's theorem states that a graph is planar if and only if it has neither K 5 nor K 3,3 as a minor. In other words, the set {K 5, K 3,3} is an obstruction set for the set of all planar graphs, and in fact the unique minimal obstruction set. A similar theorem states that K 4 and K 2,3 are the forbidden minors for the set of outerplanar graphs.