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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]
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 ...
However since T is a minimum spanning tree then T − f + e has the same weight as T, otherwise we get a contradiction and T would not be a minimum spanning tree. So T − f + e is a minimum spanning tree containing F + e and again P holds. Therefore, by the principle of induction, P holds when F has become a spanning tree, which is only ...
These algorithms find the minimum spanning forest in a possibly disconnected graph; in contrast, the most basic form of Prim's algorithm only finds minimum spanning trees in connected graphs. However, running Prim's algorithm separately for each connected component of the graph, it can also be used to find the minimum spanning forest. [9]
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.
A faster randomized minimum spanning tree algorithm based in part on Borůvka's algorithm due to Karger, Klein, and Tarjan runs in expected O(E) time. [9] The best known (deterministic) minimum spanning tree algorithm by Bernard Chazelle is also based in part on Borůvka's and runs in O(E α(E,V)) time, where α is the inverse Ackermann ...
Other optimization problems on spanning trees have also been studied, including the maximum spanning tree, the minimum tree that spans at least k vertices, the spanning tree with the fewest edges per vertex, the spanning tree with the largest number of leaves, the spanning tree with the fewest leaves (closely related to the Hamiltonian path ...
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 ...