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In graph theory and theoretical computer science, a maximum common induced subgraph of two graphs G and H is a graph that is an induced subgraph of both G and H, and that has as many vertices as possible. Finding this graph is NP-hard. In the associated decision problem, the input is two graphs G and H and a number k.
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]
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 .
Subgraph isomorphism is a generalization of the graph isomorphism problem, which asks whether G is isomorphic to H: the answer to the graph isomorphism problem is true if and only if G and H both have the same numbers of vertices and edges and the subgraph isomorphism problem for G and H is true. However the complexity-theoretic status of graph ...
Similarly, if two graphs G and H are not 2-colorable, that is, not bipartite, then both contain a cycle of odd length. Since the product of two odd cycle graphs contains an odd cycle, the product G × H is not 2-colorable either. In other words, if G × H is 2-colorable, then at least one of G and H must be 2-colorable as well.
Two graphs that have the same deck are said to be hypomorphic. With these definitions, the conjecture can be stated as: Reconstruction Conjecture: Any two hypomorphic graphs on at least three vertices are isomorphic. (The requirement that the graphs have at least three vertices is necessary because both graphs on two vertices have the same decks.)
Binary operations create a new graph from two initial graphs G 1 = (V 1, E 1) and G 2 = (V 2, E 2), such as: graph union: G 1 ∪ G 2. There are two definitions. In the most common one, the disjoint union of graphs, the union is assumed to be disjoint. Less commonly (though more consistent with the general definition of union in mathematics ...
Given a graph G and given a set L(v) of colors for each vertex v (called a list), a list coloring is a choice function that maps every vertex v to a color in the list L(v).As with graph coloring, a list coloring is generally assumed to be proper, meaning no two adjacent vertices receive the same color.