In the
mathematical field of
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, graph operations are
operations which produce new
graphs from initial ones. They include both
unary (one input) and
binary (two input) operations.
Unary operations
Unary operations create a new graph from a single initial graph.
Elementary operations
Elementary operations or editing operations, which are also known as graph edit 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 In mathematics and computer science, graph edit distance (GED) is a measure of similarity (or dissimilarity) between two graphs.
The concept of graph edit distance was first formalized mathematically by Alberto Sanfeliu and King-Sun Fu in 1983.
A m ...
between a pair of graphs is the minimum number of elementary operations required to transform one graph into the other.
Advanced operations
Advanced operations create a new graph from initial one by a complex changes, such as:
*
transpose graph
In the mathematical and algorithmic study of graph theory, the converse, transpose or reverse, entry 2.24 of a directed graph is another directed graph on the same set of vertices with all of the edges reversed compared to the orientation of t ...
;
*
complement graph
In the mathematical field of graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of a ...
;
*
line graph;
*
graph minor
In graph theory, an undirected graph is called a minor of the graph if can be formed from by deleting edges and vertices and by contracting edges.
The theory of graph minors began with Wagner's theorem that a graph is planar if and only if ...
;
*
graph rewriting;
*
power of graph
In graph theory, a branch of mathematics, the th power of an undirected graph is another graph that has the same set of vertices, but in which two vertices are adjacent when their distance in is at most . Powers of graphs are referred to ...
;
*
dual graph;
*
medial graph;
*
quotient graph In graph theory, a quotient graph ''Q'' of a graph ''G'' is a graph whose vertices are blocks of a partition of the vertices of ''G'' and where block ''B'' is adjacent to block ''C'' if some vertex in ''B'' is adjacent to some vertex in ''C'' with ...
;
*
Y-Δ transform;
*
Mycielskian
In the mathematical area of graph theory, the Mycielskian or Mycielski graph of an undirected graph is a larger graph formed from it by a construction of . The construction preserves the property of being triangle-free but increases the chromatic ...
.
Binary operations
Binary operations create a new graph from two initial graphs and , such as:
* graph union: . 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) the union of two graphs is defined as the graph .
* graph intersection: ;
* 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);
[
* graph products based on the ]cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
of the vertex sets:
** cartesian graph product: it is a commutative and associative operation (for unlabelled graphs),[ Harary, F. ''Graph Theory''. Reading, MA: Addison-Wesley, 1994.]
** lexicographic graph product (or graph composition): it is an associative (for unlabelled graphs) and non-commutative operation,[
** ]strong graph product
In graph theory, the strong product is a way of combining two graphs to make a larger graph. Two vertices are adjacent in the strong product when they come from pairs of vertices in the factor graphs that are either adjacent or identical. The str ...
: it is a commutative and associative operation (for unlabelled graphs),
** tensor graph product (or direct graph product, categorical graph product, cardinal graph product, Kronecker graph product): it is a commutative and associative operation (for unlabelled graphs),
** zig-zag graph product;
* graph product based on other products:
** rooted graph product: it is an associative operation (for unlabelled but rooted graphs),
** corona graph product: it is a non-commutative operation;Hajós construction
In graph theory, a branch of mathematics, the Hajós construction is an operation on graphs named after that may be used to construct any critical graph or any graph whose chromatic number is at least some given threshold.
The construction
Let ...
.
Notes
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