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Date: 3-8-2016
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Date: 8-5-2022
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The union of graphs
and
with disjoint point sets
and
and edge sets
and
is the graph with
and
(Harary 1994, p. 21; Gross and Yellen 2006, p. 85).
When the vertices and edges of and
are considered distinct regardless of their labels, the operation is sometimes known as the graph disjoint union in order to distinguish it from the graph union operation that merges vertices and edges with shared labels when taking the unions of edges and vertices in
and
.
The Wolfram Language function GraphUnion[g1, g2] takes the graph union by merging labeled vertices and edges, while GraphDisjointUnion[g1, g2, ...] treats vertices and edges in the components as distinct regardless of their labels.
The graph disjoint union of copies of a graph
is commonly denoted
(Harary 1990, p. 21).
Gross, J. T. and Yellen, J. Graph Theory and Its Applications, 2nd ed. Boca Raton, FL: CRC Press, 2006.
Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 21, 1994.
Skiena, S. "Unions and Intersections." §4.1.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 129-131, 1990.
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