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It follows that a non-trivial regular two-graph has an even number of points. If G is a regular graph whose two-graph extension is Γ having n points, then Γ is a regular two-graph if and only if G is a strongly regular graph with eigenvalues k, r and s satisfying n = 2(k - r) or n = 2(k - s). [9]
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.
An algebraic dual of a connected graph G is a graph G * such that G and G * have the same set of edges, any cycle of G is a cut of G *, and any cut of G is a cycle of G *. Every planar graph has an algebraic dual, which is in general not unique (any dual defined by a plane embedding will do).
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]
In graph theory, two graphs and ′ are homeomorphic if there is a graph isomorphism from some subdivision of to some subdivision of ′.If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in diagrams), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if their diagrams are homeomorphic in the ...
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 ...
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.
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.