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In the mathematical field of graph theory a graph homomorphism is a mapping between two graphs that respects their structure. More concretely it maps Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
A diagram of a graph with 6 vertices and 7 edges. ...
- vertices to vertices
- undirected edges to undirected edges or collapses the edge onto a vertex.
- directed edges to directed edges (without changing direction) or collapses the edge onto a vertex
Definition A graph homomorphism f from a graph G: = (V,E) to a graph G': = (V',E') is a function on the edges and vertices of G such that - vertices of G go to vertices of G',
- if e is an edge of G with endpoints v and w then either f(e) is an edge of G' with endpoints f(v) and f(w), or f(e) = f(v) = f(w), and
- if e is a directed edge of G from v to w then either f(e) is a directed edge of H from f(v) to f(w), or f(e) = f(v) = f(w).
The above definition works even when G and G' are allowed to have multiedges and loops. In the case of simple graphs, the definition can is slightly simpler: where an edge maps is determined by where its endpoints map. This article just presents the basic definitions. ...
Some authors use a stricter definition than the one given here, in which an edge is not allowed to map to a vertex. Thus, if the destination graph has no loops, adjacent vertices can't map to the same vertex.
If the homomorphism f is a bijection, then the inverse function is also a graph homomorphism, so f is a graph isomorphism. In this case, the two graphs are identical from the viewpoint of graph theory. Determining whether there is an isomorphism between two graphs is an important problem in computational complexity theory; see graph isomorphism problem. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...
In computational complexity theory, the graph isomorphism problem or GI problem is the graph theory problem of determining whether, given two graphs G1 and G2, it is possible to permute (or relabel) the vertices of one graph so that it is equal to the other. ...
Examples The function mapping a graph G to the complete graph with one vertex is a graph homomorphism. In the mathematical field of graph theory a complete graph is a simple graph where an edge connects every pair of vertices. ...
Notes In terms of graph coloring, a k-coloring of G, without restrictions, is equivalent to a homomorphism of G into Kk, the complete graph on k vertices. (Each vertex of G is colored according to which vertex of Kk it goes to.) As an extension of that analogy, a homomorphism of G into H is also sometimes called an H-coloring. A 3-coloring suits this graph, but fewer colors would result in adjacent vertices of the same color. ...
In the mathematical field of graph theory a complete graph is a simple graph where an edge connects every pair of vertices. ...
Properties In mathematics and computer science the connectivity of graphs is one of the basic concepts of graph theory. ...
One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ...
One major problem that has plagued graph theory since its inception is the consistent lack of consistency in terminology. ...
In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. ...
In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ...
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