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Minor testing (checking whether an input graph contains an input graph as a minor); the same holds with topological minors; Steiner tree, or Minimum spanning tree for a subset of the vertices of a graph. [2] (The minimum spanning tree for an entire graph is solvable in polynomial time.) Modularity maximization [5]
with a corresponding factor graph shown on the right. Observe that the factor graph has a cycle. If we merge (,) (,) into a single factor, the resulting factor graph will be a tree. This is an important distinction, as message passing algorithms are usually exact for trees, but only approximate for graphs with cycles.
If a connected graph is a Cartesian product, it can be factorized uniquely as a product of prime factors, graphs that cannot themselves be decomposed as products of graphs. [2] However, Imrich & Klavžar (2000) describe a disconnected graph that can be expressed in two different ways as a Cartesian product of prime graphs:
In graph theory, a graph product is a binary operation on graphs. Specifically, it is an operation that takes two graphs G 1 and G 2 and produces a graph H with the following properties: The vertex set of H is the Cartesian product V ( G 1 ) × V ( G 2 ) , where V ( G 1 ) and V ( G 2 ) are the vertex sets of G 1 and G 2 , respectively.
Each tree node lists at most three vertices, so the width of this decomposition is two. In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph. Tree decompositions are also called junction trees, clique trees ...
Cayley Q8 graph of quaternion multiplication showing cycles of multiplication of i (red), j (green) and k (blue). In the SVG file, hover over or click a path to highlight it. The next step in the construction is to generalize the multiplication and conjugation operations. Form ordered pairs (a, b) of complex numbers a and b, with multiplication ...
Other problems specify a family of graphs into which a given graph should be decomposed, for instance, a family of cycles, or decomposing a complete graph K n into n − 1 specified trees having, respectively, 1, 2, 3, ..., n − 1 edges. Some specific decomposition problems that have been studied include:
An odd number does not have the prime factor 2. The first ... A primorial x# is the product of all primes from 2 to x. The first: 2, 6, 30, 210 ... 62: 2·31 63: 3 2 ...