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  2. Planar graph - Wikipedia

    en.wikipedia.org/wiki/Planar_graph

    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.

  3. Complete graph - Wikipedia

    en.wikipedia.org/wiki/Complete_graph

    The complement graph of a complete graph is an empty graph. If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. K n can be decomposed into n trees T i such that T i has i vertices. [6] Ringel's conjecture asks if the complete graph K 2n+1 can be decomposed into copies of any tree ...

  4. Mac Lane's planarity criterion - Wikipedia

    en.wikipedia.org/wiki/Mac_Lane's_planarity_criterion

    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 ...

  5. Whitney's planarity criterion - Wikipedia

    en.wikipedia.org/wiki/Whitney's_planarity_criterion

    A planar graph and its dual. Every cycle in the blue graph is a minimal cut in the red graph, and vice versa, so the two graphs are algebraic duals and have dual graphic matroids. In mathematics, Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney. [1]

  6. Clique problem - Wikipedia

    en.wikipedia.org/wiki/Clique_problem

    For graphs of constant arboricity, such as planar graphs (or in general graphs from any non-trivial minor-closed graph family), this algorithm takes O (m) time, which is optimal since it is linear in the size of the input. [18] If one desires only a single triangle, or an assurance that the graph is triangle-free, faster algorithms are possible.

  7. Kuratowski's theorem - Wikipedia

    en.wikipedia.org/wiki/Kuratowski's_theorem

    The two graphs and , are nonplanar, as may be shown either by a case analysis or an argument involving Euler's formula. Additionally, subdividing a graph cannot turn a nonplanar graph into a planar graph: if a subdivision of a graph has a planar drawing, the paths of the subdivision form curves that may be used to represent the edges of itself ...

  8. Grinberg's theorem - Wikipedia

    en.wikipedia.org/wiki/Grinberg's_theorem

    The proof is an easy consequence of Euler's formula. [1] [2] As a corollary of this theorem, if an embedded planar graph has only one face whose number of sides is not 2 mod 3, and the remaining faces all have numbers of sides that are 2 mod 3, then the graph is not Hamiltonian.

  9. Cycle basis - Wikipedia

    en.wikipedia.org/wiki/Cycle_basis

    Alternatively, if the edges of the graph have positive weights, the minimum weight cycle basis may be constructed in polynomial time. In planar graphs, the set of bounded cycles of an embedding of the graph forms a cycle basis. The minimum weight cycle basis of a planar graph corresponds to the Gomory–Hu tree of the dual graph.