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If a graph G has a linkless or flat embedding, then every minor of G (a graph formed by contraction of edges and deletion of edges and vertices) also has a linkless or flat embedding. Deletions cannot destroy the flatness of an embedding, and a contraction can be performed by leaving one endpoint of the contracted edge in place and rerouting ...
Switching {X,Y} in a graph. A two-graph is equivalent to a switching class of graphs and also to a (signed) switching class of signed complete graphs.. Switching a set of vertices in a (simple) graph means reversing the adjacencies of each pair of vertices, one in the set and the other not in the set: thus the edge set is changed so that an adjacent pair becomes nonadjacent and a nonadjacent ...
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.
A graph isomorphism is a one-to-one incidence preserving correspondence of the vertices and edges of one graph to the vertices and edges of another graph. Two graphs related in this way are said to be isomorphic. isoperimetric See expansion. isthmus Synonym for bridge, in the sense of an edge whose removal disconnects the graph.
Clique-sums are important in graph structure theory, where they are used to characterize certain families of graphs as the graphs formed by clique-sums of simpler graphs. The first result of this type [2] was a theorem of Wagner (1937), who proved that the graphs that do not have a five-vertex complete graph as a minor are the 3-clique-sums of ...
The Cartesian product of two path graphs is a grid graph. The Cartesian product of n edges is a hypercube: =. Thus, the Cartesian product of two hypercube graphs is another hypercube: Q i Q j = Q i+j. The Cartesian product of two median graphs is another median graph. The graph of vertices and edges of an n-prism is the Cartesian product graph ...
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Two graphs G and H are homomorphically equivalent if G → H and H → G. [4] The maps are not necessarily surjective nor injective. For instance, the complete bipartite graphs K 2,2 and K 3,3 are homomorphically equivalent: each map can be defined as taking the left (resp. right) half of the domain graph and mapping to just one vertex in the left (resp. right) half of the image graph.