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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.
While graph drawing and graph representation are valid topics in graph theory, in order to focus only on the abstract structure of graphs, a graph property is defined to be a property preserved under all possible isomorphisms of a graph. In other words, it is a property of the graph itself, not of a specific drawing or representation of the graph.
In graph theory, the Weisfeiler Leman graph isomorphism test is a heuristic test for the existence of an isomorphism between two graphs G and H. [1] It is a generalization of the color refinement algorithm and has been first described by Weisfeiler and Leman in 1968. [ 2 ]
A set of graphs isomorphic to each other is called an isomorphism class of graphs. The question of whether graph isomorphism can be determined in polynomial time is a major unsolved problem in computer science, known as the graph isomorphism problem. [1] [2] The two graphs shown below are isomorphic, despite their different looking drawings.
Two graphs are isomorphic if there is an isomorphism between them; see isomorphism. isomorphism 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
The special case of finding a long path as an induced subgraph of a hypercube has been particularly well-studied, and is called the snake-in-the-box problem. [3] The maximum independent set problem is also an induced subgraph isomorphism problem in which one seeks to find a large independent set as an induced subgraph of a larger graph, and the maximum clique problem is an induced subgraph ...
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.)
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.