In the mathematics|mathematical field of graph theory, a spanning tree T of a connected graph|connected, undirected graph G is a tree (graph theory)|tree composed of all the Vertex (graph theory)|vertices and some (or perhaps all) of the Edge (graph theory)|edges of G. Informally, a spanning tree of G is a selection of edges of G that form a tree spanning every vertex. That is, every vertex lies in the tree, but no cycle (graph theory)|cycles (or loops) are formed.
On the other hand, every bridge (graph theory)|bridge of G must belong to T.
A spanning tree of a connected graph G can also be defined as a maximal set of edges of G that contains no cycle, or as a minimal set of edges that connect all vertices.
In certain fields of graph theory it is often useful to find a minimum spanning tree of a weighted graph. 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 minimum spanning tree with at most k edges per vertex degree-constrained spanning tree, the spanning tree with the largest number of leaves (closely related to the smallest connected dominating set), the spanning tree with the fewest leaves (closely related to the Hamiltonian path problem), the minimum diameter spanning tree, and the minimum dilation spanning tree.
In the mathematics|mathematical field of graph theory, a spanning tree T of a connected graph|connected, undirected graph G is a tree (graph theory)|tree composed of all the Vertex (graph theory)|vertices and some (or perhaps all) of the Edge (graph theory)|edges of G. Informally, a spanning tree of G is a selection of edges of G that form a tree spanning every vertex. That is, every vertex lies in the tree, but no cycle (graph theory)|cycles (or loops) are formed.
On the other hand, every bridge (graph theory)|bridge of G must belong to T.
A spanning tree of a connected graph G can also be defined as a maximal set of edges of G that contains no cycle, or as a minimal set of edges that connect all vertices.
In certain fields of graph theory it is often useful to find a minimum spanning tree of a weighted graph. 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 minimum spanning...