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In particular, we can adjust it to merge (link) and split (cut) in O(log(n)) amortized time. 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 ...
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 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.
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]
Label each split component with a P (a two-vertex split component with multiple edges), an S (a split component in the form of a triangle), or an R (any other split component). While there exist two split components that share a linked pair of virtual edges, and both components have type S or both have type P, merge them into a single larger ...
Create a forest (a set of trees) initially consisting of a separate single-vertex tree for each vertex in the input graph. Sort the graph edges by weight. Loop through the edges of the graph, in ascending sorted order by their weight. For each edge: Test whether adding the edge to the current forest would create a cycle.
The vertex set of X is the set of cosets VX = {gB : g ∈ G}. Two vertices gB and fB are adjacent in X whenever there exists b in B such that either fB = gbeB or fB = gbe −1 B. The G-stabilizer of every vertex of X is conjugate to B in G and the stabilizer of every edge of X is conjugate to H in G. Every vertex of X has degree equal to [B : H ...
The two queries partition the vertex set into 4 subsets: vertices reached by both, either one, or none of the searches. One can show that a strongly connected component has to be contained in one of the subsets. The vertex subset reached by both searches forms a strongly connected component, and the algorithm then recurses on the other 3 subsets.