# Graph operations

(Redirected from Operations on graphs)

Graph operations produce new graphs from initial ones. They may be separated into the following major categories.

## Unary operations

Unary operations create a new graph from one initial one.

### Elementary operations

Elementary operations or editing operations create a new graph from one initial one by a simple local change, such as addition or deletion of a vertex or of an edge, merging and splitting of vertices, edge contraction, etc. The graph edit distance between a pair of graphs is the minimum number of elementary operations required to transform one graph into the other.

Advanced operations create a new graph from one initial one by a complex changes, such as:

## Binary operations

Binary operations create a new graph from two initial ones G1 = (V1, E1) and G2 = (V2, E2), such as:

• graph union: G1G2 = (V1V2, E1E2). When V1 and V2 are disjoint, the graph union is referred to as the disjoint graph union, and denoted G1G2;[1]
• graph intersection: G1G2 = (V1V2, E1E2);[1]
• graph join: graph with all the edges that connect the vertices of the first graph with the vertices of the second graph. It is a commutative operation (for unlabelled graphs);[2]
• graph products based on the cartesian product of the vertex sets:
• graph product based on other products:
• series-parallel graph composition:
• parallel graph composition: it is a commutative operation (for unlabelled graphs),
• series graph composition: it is a non-commutative operation,
• source graph composition: it is a commutative operation (for unlabelled graphs);
• Hajós construction.

## Notes

1. ^ a b Bondy, J. A.; Murty, U. S. R. (2008). Graph Theory. Graduate Texts in Mathematics. Springer. p. 29. ISBN 978-1-84628-969-9.
2. ^ a b c Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.
3. ^ Reingold, O.; Vadhan, S.; Wigderson, A. (2002). "Entropy waves, the zig-zag graph product, and new constant-degree expanders". Annals of Mathematics. 155 (1): 157–187. doi:10.2307/3062153. JSTOR 3062153. MR 1888797.
4. ^ Robert Frucht and Frank Harary. "On the corona of two graphs", Aequationes Math., 4:322–324, 1970.