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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).
Minimum k-spanning tree; Minor testing (checking whether an input graph contains an input graph as a minor); the same holds with topological minors; 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.)
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] The graph on the right is ...
Zorn's lemma can be used to show that every connected graph has a spanning tree. The set of all sub-graphs that are trees is ordered by inclusion, and the union of a chain is an upper bound. Zorn's lemma says that a maximal tree must exist, which is a spanning tree since the graph is connected. [1] Zorn's lemma is not needed for finite graphs ...
A complete bipartite graph K m,n has m n−1 n m−1 spanning trees. [13] A complete bipartite graph K m,n has a maximum matching of size min{m,n}. A complete bipartite graph K n,n has a proper n-edge-coloring corresponding to a Latin square. [14]
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 coloring number of a graph, also known as its Szekeres-Wilf number (Szekeres & Wilf 1968) is always equal to its degeneracy plus 1 (Jensen & Toft 1995, p. 77f.). The strength of a graph is a fractional value whose integer part gives the maximum number of disjoint spanning trees that can be drawn in a graph. It is the packing problem that is ...
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]