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  2. Gallai–Edmonds decomposition - Wikipedia

    en.wikipedia.org/wiki/Gallai–Edmonds_decomposition

    In graph theory, the Gallai–Edmonds decomposition is a partition of the vertices of a graph into three subsets which provides information on the structure of maximum matchings in the graph. Tibor Gallai [1] [2] and Jack Edmonds [3] independently discovered it and proved its key properties. The Gallai–Edmonds decomposition of a graph can be ...

  3. 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.

  4. Pathwidth - Wikipedia

    en.wikipedia.org/wiki/Pathwidth

    One way to find a path-decomposition with this width is (similarly to the logarithmic-width path-decomposition of forests described above) to use the planar separator theorem to find a set of O(√ n) vertices the removal of which separates the graph into two subgraphs of at most 2n ⁄ 3 vertices each, and concatenate recursively-constructed ...

  5. Split (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Split_(graph_theory)

    In graph theory, a split of an undirected graph is a cut whose cut-set forms a complete bipartite graph.A graph is prime if it has no splits. The splits of a graph can be collected into a tree-like structure called the split decomposition or join decomposition, which can be constructed in linear time.

  6. Graph theory - Wikipedia

    en.wikipedia.org/wiki/Graph_theory

    The first textbook on graph theory was written by Dénes Kőnig, and published in 1936. [26] Another book by Frank Harary , published in 1969, was "considered the world over to be the definitive textbook on the subject", [ 27 ] and enabled mathematicians, chemists, electrical engineers and social scientists to talk to each other.

  7. (a, b)-decomposition - Wikipedia

    en.wikipedia.org/wiki/(a,_b)-decomposition

    In graph theory, the (a, b)-decomposition of an undirected graph is a partition of its edges into a + 1 sets, each one of them inducing a forest, except one which induces a graph with maximum degree b. If this graph is also a forest, then we call this a F(a, b)-decomposition. A graph with arboricity a is (a, 0)-decomposable.

  8. Modular decomposition - Wikipedia

    en.wikipedia.org/wiki/Modular_decomposition

    The decomposition depicted in the figure below is this special decomposition for the given graph. A graph, its quotient where "bags" of vertices of the graph correspond to the children of the root of the modular decomposition tree, and its full modular decomposition tree: series nodes are labeled "s", parallel nodes "//" and prime nodes "p".

  9. Branch-decomposition - Wikipedia

    en.wikipedia.org/wiki/Branch-decomposition

    Branch decomposition of a grid graph, showing an e-separation.The separation, the decomposition, and the graph all have width three. In graph theory, a branch-decomposition of an undirected graph G is a hierarchical clustering of the edges of G, represented by an unrooted binary tree T with the edges of G as its leaves.