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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.
Geometrically K 3 forms the edge set of a triangle, K 4 a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K 7 as its skeleton. [15] Every neighborly polytope in four or more dimensions also has a complete skeleton. K 1 through K 4 are all planar graphs.
However, some bounded-treewidth planar graphs such as the nested triangles graph may be -outerplanar only for very large , linear in the number of vertices. Baker's technique covers a planar graph with a constant number of k {\displaystyle k} -outerplanar graphs and uses their low treewidth in order to quickly approximate several hard graph ...
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 ...
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 ...
A 3-coloring of a triangle-free planar graph. In the mathematical field of graph theory, Grötzsch's theorem is the statement that every triangle-free planar graph can be colored with only three colors.
K 5 and K 3,3: Graph minor Wagner's theorem: Outerplanar graphs: K 4 and K 2,3: Graph minor Diestel (2000), [1] p. 107: Outer 1-planar graphs: Six forbidden minors Graph minor Auer et al. (2013) [2] Graphs of fixed genus: A finite obstruction set Graph minor Diestel (2000), [1] p. 275: Apex graphs: A finite obstruction set Graph minor [3 ...