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  2. Two-graph - Wikipedia

    en.wikipedia.org/wiki/Two-graph

    This two-graph is a regular two-graph since each pair of distinct vertices appears together in exactly two triples. Given a simple graph G = (V,E), the set of triples of the vertex set V whose induced subgraph has an odd number of edges forms a two-graph on the set V. Every two-graph can be represented in this way. [1]

  3. Bicircular matroid - Wikipedia

    en.wikipedia.org/wiki/Bicircular_matroid

    The closed sets (flats) of the bicircular matroid of a graph G can be described as the forests F of G such that in the induced subgraph of V(G) − V(F), every connected component has a cycle. Since the flats of a matroid form a geometric lattice when partially ordered by set inclusion, these forests of G also form a geometric

  4. Whitney's planarity criterion - Wikipedia

    en.wikipedia.org/wiki/Whitney's_planarity_criterion

    An equivalent form of Whitney's criterion is that a graph G is planar if and only if it has a dual graph whose graphic matroid is dual to the graphic matroid of G. A graph whose graphic matroid is dual to the graphic matroid of G is known as an algebraic dual of G. Thus, Whitney's planarity criterion can be expressed succinctly as: a graph is ...

  5. Graphic matroid - Wikipedia

    en.wikipedia.org/wiki/Graphic_matroid

    Some classes of matroid have been defined from well-known families of graphs, by phrasing a characterization of these graphs in terms that make sense more generally for matroids. These include the bipartite matroids , in which every circuit is even, and the Eulerian matroids , which can be partitioned into disjoint circuits.

  6. Dual graph - Wikipedia

    en.wikipedia.org/wiki/Dual_graph

    A connected graph G is planar if and only if it has an algebraic dual. The same fact can be expressed in the theory of matroids. If M is the graphic matroid of a graph G, then a graph G * is an algebraic dual of G if and only if the graphic matroid of G * is the dual matroid of M.

  7. Planar graph - Wikipedia

    en.wikipedia.org/wiki/Planar_graph

    The above is a direct corollary of the fact that a graph G is outerplanar if the graph formed from G by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph. [8] A 1-outerplanar embedding of a graph is the same as an outerplanar embedding.

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  9. Linkless embedding - Wikipedia

    en.wikipedia.org/wiki/Linkless_embedding

    A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three-dimensional analogue of the planar graphs. [1] Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding. Flat embeddings are automatically linkless, but not vice versa. [2]