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The canonical form of a graph is an example of a complete graph invariant: every two isomorphic graphs have the same canonical form, and every two non-isomorphic graphs have different canonical forms. [1] [2] Conversely, every complete invariant of graphs may be used to construct a canonical form. [3]
A canonical form is a labeled graph Canon(G) that is isomorphic to G, such that every graph that is isomorphic to G has the same canonical form as G. Thus, from a solution to the graph canonization problem, one could also solve the problem of graph isomorphism : to test whether two graphs G and H are isomorphic, compute their canonical forms ...
The original formulation is based on graph canonization, a normal form for graphs, while there is also a combinatorial interpretation in the spirit of color refinement and a connection to logic. There are several versions of the test (e.g. k-WL and k-FWL) referred to in the literature by various names, which easily leads to confusion.
In general, a graph may have multiple double covers that are different from the bipartite double cover. [6] The graph C is a covering graph of H if there is a surjective local isomorphism f from C to H. In the figure, the surjection is indicated by the colours. For example, f maps both blue nodes in C to the blue node in H.
The Jordan form is used to find a normal form of matrices up to conjugacy such that normal matrices make up an algebraic variety of a low fixed degree in the ambient matrix space. Sets of representatives of matrix conjugacy classes for Jordan normal form or rational canonical forms in general do not constitute linear or affine subspaces in the ...
Canonical forms may also be called canonical invariants or complete invariants, and are sometimes defined only for the graphs within a particular family of graphs. Graph canonization is the process of computing a canonical form. card A graph formed from a given graph by deleting one vertex, especially in the context of the reconstruction ...
In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata.These automata represent the Cayley graph of the group. That is, they can tell whether a given word representation of a group element is in a "canonical form" and can tell whether two elements given in canonical words differ by a generator.
Historically, the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, and is in turn generalized combinatorially by the concept of a dual matroid.