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A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. [ 2 ] A directed tree, [ 3 ] oriented tree, [ 4 ] [ 5 ] polytree , [ 6 ] or singly connected network [ 7 ] is a directed acyclic graph (DAG) whose underlying ...
Trees are commonly used to represent or manipulate hierarchical data in applications such as: . File systems for: . Directory structure used to organize subdirectories and files (symbolic links create non-tree graphs, as do multiple hard links to the same file or directory)
To traverse arbitrary trees (not necessarily binary trees) with depth-first search, perform the following operations at each node: If the current node is empty then return. Visit the current node for pre-order traversal. For each i from 1 to the current node's number of subtrees − 1, or from the latter to the former for reverse traversal, do:
With the SPQR tree of a graph G (without Q nodes) it is straightforward to find every pair of vertices u and v in G such that removing u and v from G leaves a disconnected graph, and the connected components of the remaining graphs: The two vertices u and v may be the two endpoints of a virtual edge in the graph associated with an R node, in ...
A naive approach might be to build two parallel trees, one ordered by the beginning point, and one ordered by the ending point of each interval. This allows discarding half of each tree in O ( log n ) {\displaystyle O(\log n)} time, but the results must be merged, requiring O ( n ) {\displaystyle O(n)} time.
For simple connected graphs, shortest-path trees can be used [1] to suggest a non-linear relationship between two network centrality measures, closeness and degree. By assuming that the branches of the shortest-path trees are statistically similar for any root node in one network, one may show that the size of the branches depend only on the ...
A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic. A polytree is an example of an oriented graph. The term polytree was coined in 1987 by Rebane and ...
K-trees are chordal graphs in which all maximal cliques and all maximal clique separators have the same size. [10] Apollonian networks are chordal maximal planar graphs, or equivalently planar 3-trees. [10] Maximal outerplanar graphs are a subclass of 2-trees, and therefore are also chordal.