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It has n 2 + 2n + 1 vertices: n 2 formed from the product of a leaf in both factors, 2n from the product of a leaf in one factor and the hub in the other factor, and one remaining vertex formed from the product of the two hubs. Each leaf-hub product vertex in G dominates exactly n of the leaf-leaf vertices, so n leaf-hub vertices are needed to ...
Simultaneous embedding is a technique in graph drawing and information visualization for visualizing two or more different graphs on the same or overlapping sets of labeled vertices, while avoiding crossings within both graphs. Crossings between an edge of one graph and an edge of the other graph are allowed. [1]
The modular product of graphs. In graph theory, the modular product of graphs G and H is a graph formed by combining G and H that has applications to subgraph isomorphism.It is one of several different kinds of graph products that have been studied, generally using the same vertex set (the Cartesian product of the sets of vertices of the two graphs G and H) but with different rules for ...
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.
Edge colorings are invariant to amalgamation. This is obvious, as all of the edges between the two graphs are in bijection with each other. However, what may not be obvious, is that if is a complete graph of the form +, and we color the edges as to specify a Hamiltonian decomposition (a decomposition into Hamiltonian paths, then those edges also form a Hamiltonian Decomposition in .
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 ...
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Let G and H be any two k-constructible graphs. Then the graph formed by applying the Hajós construction to G and H is k-constructible. Let G be any k-constructible graph, and let u and v be any two non-adjacent vertices in G. Then the graph formed by combining u and v into a single vertex is also k-constructible.