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. 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). If all of the edges of G are also edges of a spanning tree T of G, then G is a tree and is identical to T (that is, a tree has a unique spanning tree and it is itself).
Applications Several pathfinding algorithms, including Dijkstra's algorithm and the A* search algorithm, internally build a spanning tree as an intermediate step in solving the problem. In order to minimize the cost of power networks, wiring connections, piping, automatic speech recognition, etc., people often use algorithms that gradually build a spanning tree (or many such trees) as intermediate steps in the process of finding the minimum spanning tree. The Internet and many other telecommunications networks have transmission links that connect nodes together in a mesh topology that includes some loops. In order to avoid bridge loops and routing loops, many routing protocols designed for such networks—including the Spanning Tree Protocol, Open Shortest Path First, Link-state routing protocol, Augmented tree-based routing, etc.—require each router to remember a spanning tree. A special kind of spanning tree, the Xuong tree, is used in topological graph theory to find graph embeddings with maximum genus. A Xuong tree is a spanning tree such that, in the remaining graph, the number of connected components with an odd number of edges is as small as possible. A Xuong tree and an associated maximum-genus embedding can be found in polynomial time.
Definitions A tree is a connected undirected graph with no cycles. It is a spanning tree of a graph G if it spans G (that is, it includes every vertex of G) and is a subgraph of G (every edge in the tree belongs to G). 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.
Fundamental cycles Adding just one edge to a spanning tree will create a cycle; such a cycle is called a fundamental cycle with respect to that tree. There is a distinct fundamental cycle for each edge not in the spanning tree; thus, there is a one-to-one correspondence between fundamental cycles and edges not in the spanning tree. For a connected graph with V vertices, any spanning tree will have V − 1 edges, and thus, a graph of E edges and one of its spanning trees will have E − V + 1 fundamental cycles (The number of edges subtracted by number of edges included in a spanning tree; giving the number of edges not included in the spanning tree). For any given spanning tree the set of all E − V + 1 fundamental cycles forms a cycle basis, i.e., a basis for the cycle space.
Fundamental cutsets Dual to the notion of a fundamental cycle is the notion of a fundamental cutset with respect to a given spanning tree. By deleting just one edge of the spanning tree, the vertices are partitioned into two disjoint sets. The fundamental cutset is defined as the set of edges that must be removed from the graph G to accomplish the same partition. Thus, each spanning tree defines a set of V − 1 fundamental cutsets, one for each edge of the spanning tree. The duality between fundamental cutsets and fundamental cycles is established by noting that cycle edges not in the spanning tree can only appear in the cutsets of the other edges in the cycle; and vice versa: edges in a cutset can only appear in those cycles containing the edge corresponding to the cutset. This duality can also be expressed using the theory of matroids, according to which a spanning tree is a base of the graphic matroid, a fundamental cycle is the unique circuit within the set formed by adding one element to the base, and fundamental cutsets are defined in the same way from the dual matroid.
Spanning forests A collection of disjoint (unconnected) trees is described as a forest. A spanning forest in a graph is a subgraph that is a forest with an additional requirement. There are two incompatible requirements in use, of which one is relatively rare.
Almost all graph theory books and articles define a spanning forest as a forest that spans all of the vertices, meaning only that each vertex of the graph is a vertex in the forest. A connected graph may have a disconnected spanning forest, such as the forest with no edges, in which each vertex forms a single-vertex tree. A few graph theory authors define a spanning forest to be a maximal acyclic subgraph of the given graph, or equivalently a subgraph consisting of a spanning tree in each connected component of the graph. To avoid confusion between these two definitions, Gross & Yellen (2005) suggest the term "full spanning forest" for a spanning forest with the same number of components as the given graph (i.e., a maximal forest), while Bondy & Murty (2008) instead call this kind of forest a "maximal spanning forest" (which is redundant, as a maximal forest necessarily contains every vertex).
Counting spanning trees
The number t(G) of spanning trees of a connected graph is a well-studied invariant.
In specific graphs In some cases, it is easy to calculate t(G) directly:
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