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A cut or split is trivial when one of its two sides has only one vertex in it; every trivial cut is a split. A graph is said to be prime (with respect to splits) if it has no nontrivial splits. [2] Two splits are said to cross if each side of one split has a non-empty intersection with each side of the other split.
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]
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.
In any split graph, one of the following three possibilities must be true: [10] There exists a vertex x in i such that C ∪ {x} is complete. In this case, C ∪ {x} is a maximum clique and i is a maximum independent set. There exists a vertex x in C such that i ∪ {x} is independent.
In either case, one can form a spanning tree by connecting each vertex, other than the root vertex v, to the vertex from which it was discovered. This tree is known as a depth-first search tree or a breadth-first search tree according to the graph exploration algorithm used to construct it. [ 18 ]
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.
Link/cut trees divide each tree in the represented forest into vertex-disjoint paths, where each path is represented by an auxiliary data structure (often splay trees, though the original paper predates splay trees and thus uses biased binary search trees). The nodes in the auxiliary data structure are ordered by their depth in the ...
Steiner tree problems in graphs are applied to various problems in research and industry, [7] including multicast routing [8] and bioinformatics. [9] A special case of this problem is when G is a complete graph, each vertex v ∈ V corresponds to a point in a metric space, and the edge weights w(e) for each e ∈ E correspond to distances in ...