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In the mathematical field of graph theory, a spanning tree T of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G. [1] In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree (see about spanning forests below).
In directed graphs, the maximum spanning tree solution cannot be used.Instead, several different algorithms are known; the choice of which algorithm to use depends on whether a start or destination vertex for the path is fixed, or whether paths for many start or destination vertices must be found simultaneously.
In an undirected graph G(V, E) and a function w : E → R, let S be the set of all spanning trees T i. Let B(T i) be the maximum weight edge for any spanning tree T i. We define subset of minimum bottleneck spanning trees S′ such that for every T j ∈ S′ and T k ∈ S we have B(T j) ≤ B(T k) for all i and k. [2]
Maximum Induced path [3]: GT23 Minimum maximal independent set a.k.a. minimum independent dominating set [4] NP-complete special cases include the minimum maximal matching problem, [3]: GT10 which is essentially equal to the edge dominating set problem (see above). Metric dimension of a graph [3]: GT61 Metric k-center; Minimum degree spanning tree
A planar graph and its minimum spanning tree. Each edge is labeled with its weight, which here is roughly proportional to its length. A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. [1]
On a weighted graph, a Degree-constrained minimum spanning tree (DCMST) is a degree-constrained spanning tree in which the sum of its edges has the minimum possible sum. Finding a DCMST is an NP-Hard problem. [1] Heuristic algorithms that can solve the problem in polynomial time have been proposed, including Genetic and Ant-Based Algorithms.
Edmonds' algorithm (also known as Chu–Liu/Edmonds' algorithm): find maximum or minimum branchings; Euclidean minimum spanning tree: algorithms for computing the minimum spanning tree of a set of points in the plane; Longest path problem: find a simple path of maximum length in a given graph; Minimum spanning tree. Borůvka's algorithm ...
Any spanning tree T of a graph G has at least two leaves, vertices that have only one edge of T incident to them. A maximum leaf spanning tree is a spanning tree that has the largest possible number of leaves among all spanning trees of G. The max leaf number of G is the number of leaves in the maximum leaf spanning tree. [2]