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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]
The quality of the tree is measured in the same way as in a graph, using the Euclidean distance between pairs of points as the weight for each edge. Thus, for instance, a Euclidean minimum spanning tree is the same as a graph minimum spanning tree in a complete graph with Euclidean edge weights.
A Euclidean minimum spanning tree, for a set of points in the Euclidean plane or Euclidean space, is a system of line segments, having only the given points as their endpoints, whose union includes all of the points in a connected set, and which has the minimum possible total length of any such system.
In graph theory, a minimum degree spanning tree is a subset of the edges of a connected graph that connects all the vertices together, without any cycles, and its maximum degree of its vertices as small as possible. That is, it is a spanning tree whose maximum degree is minimal.
Kruskal's algorithm [1] finds a minimum spanning forest of an undirected edge-weighted graph.If the graph is connected, it finds a minimum spanning tree.It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle. [2]
For other graphs, the expected weight of the random minimum spanning tree can be calculated as an integral involving the Tutte polynomial of the graph. [ 2 ] In contrast to uniformly random spanning trees of complete graphs, for which the typical diameter is proportional to the square root of the number of vertices, random minimum spanning ...
The shortest-path tree from this point to all vertices in the graph is a minimum-diameter spanning tree of the graph. [2] The absolute 1-center problem was introduced long before the first study of the minimum-diameter spanning tree problem, [ 2 ] [ 3 ] and in a graph with n {\displaystyle n} vertices and m {\displaystyle m} edges it can be ...
In graph theory, Edmonds' algorithm or Chu–Liu/Edmonds' algorithm is an algorithm for finding a spanning arborescence of minimum weight (sometimes called an optimum branching). It is the directed analog of the minimum spanning tree problem.