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A flat embedding is necessarily linkless, but there may exist linkless embeddings that are not flat: for instance, if G is a graph formed by two disjoint cycles, and it is embedded to form the Whitehead link, then the embedding is linkless but not flat.
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 ...
Given an embedding G of a (not necessarily simple) connected graph in the plane without edge intersections, we construct the dual graph G* as follows: we choose one vertex in each face of G (including the outer face) and for each edge e in G we introduce a new edge in G* connecting the two vertices in G* corresponding to the two faces in G that ...
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.
Other pairs of graph types that always admit a simultaneous embedding, but that might need larger grid sizes, include a wheel graph and a cycle graph, a tree and a matching, or a pair of graphs both of which have maximum degree two. However, pairs of planar graphs and a matching, or of a Angelini, Geyer, Neuwirth and Kaufmann showed that a tree ...
The rank in () of a set of edges of a graph is () = where is the number of vertices in the subgraph formed by the edges in and is the number of connected components of the same subgraph. [2] The corank of the graphic matroid is known as the circuit rank or cyclomatic number.
A 2-sum of 2-flattenable graphs is 2-flattenable if and only if at most one graph has a minor. The fact that K 4 {\displaystyle K_{4}} is 2-flattenable but K 5 {\displaystyle K_{5}} is not has implications on the forbidden minor characterization for 2-flattenable graphs under the l 1 {\displaystyle l_{1}} -norm.
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]