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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 ...
Two graphs are homomorphically equivalent if there exist two homomorphisms, one from each graph to the other graph. homomorphism 1. A graph homomorphism is a mapping from the vertex set of one graph to the vertex set of another graph that maps adjacent vertices to adjacent vertices. This type of mapping between graphs is the one that is most ...
For instance, if G and H are both connected graphs, each having at least four vertices and having exactly twice as many total vertices as their domination numbers, then γ(G H) = γ(G) γ(H). [2] The graphs G and H with this property consist of the four-vertex cycle C 4 together with the rooted products of a connected graph and a single edge. [2]
A multigraph with multiple edges (red) and several loops (blue). Not all authors allow multigraphs to have loops. In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called parallel edges [1]), that is, edges that have the same end nodes.
Clique-sums have a close connection with treewidth: If two graphs have treewidth at most k, so does their k-clique-sum.Every tree is the 1-clique-sum of its edges. Every series–parallel graph, or more generally every graph with treewidth at most two, may be formed as a 2-clique-sum of triangles.
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 .
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A graph G is said to be k-constructible (or Hajós-k-constructible) when it formed in one of the following three ways: [1] The complete graph K k is k-constructible. 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.