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A graph with eight vertices, and a tree decomposition of it onto a tree with six nodes. Each graph edge connects two vertices that are listed together at some tree node, and each graph vertex is listed at the nodes of a contiguous subtree of the tree. Each tree node lists at most three vertices, so the width of this decomposition is two.
In graph theory, a split of an undirected graph is a cut whose cut-set forms a complete bipartite graph.A graph is prime if it has no splits. The splits of a graph can be collected into a tree-like structure called the split decomposition or join decomposition, which can be constructed in linear time.
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] Monochromatic triangle [3]: GT6 Pathwidth, [6] or, equivalently, interval thickness, and vertex separation number [7] Rank coloring; k-Chinese postman
A recursive tree is a labeled rooted tree where the vertex labels respect the tree order (i.e., if u < v for two vertices u and v, then the label of u is smaller than the label of v). In a rooted tree, the parent of a vertex v is the vertex connected to v on the path to the root; every vertex has a unique parent, except the root has no parent. [24]
The left tree is the decision tree we obtain from using information gain to split the nodes and the right tree is what we obtain from using the phi function to split the nodes. The resulting tree from using information gain to split the nodes. Now assume the classification results from both trees are given using a confusion matrix.
As with finite graphs, a tree is a connected graph with no finite cycles, and a spanning tree can be defined either as a maximal acyclic set of edges or as a tree that contains every vertex. [ 27 ] The trees within a graph may be partially ordered by their subgraph relation, and any infinite chain in this partial order has an upper bound (the ...
A cutpoint, cut vertex, or articulation point of a graph G is a vertex that is shared by two or more blocks. The structure of the blocks and cutpoints of a connected graph can be described by a tree called the block-cut tree or BC-tree. This tree has a vertex for each block and for each articulation point of the given graph.
When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.