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In graph theory, the Nash-Williams theorem is a tree-packing theorem that describes how many edge-disjoint spanning trees (and more generally forests) a graph can have:. A graph G has t edge-disjoint spanning trees iff for every partition , …, where there are at least t(k − 1) crossing edges (Tutte 1961, Nash-Williams 1961).
A cut C = (S, T) is a partition of V of a graph G = (V, E) into two subsets S and T. The cut-set of a cut C = (S, T) is the set {(u, v) ∈ E | u ∈ S, v ∈ T} of edges that have one endpoint in S and the other endpoint in T. If s and t are specified vertices of the graph G, then an s – t cut is a cut in which s belongs to the set S and t ...
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An edge e in G is a bridge if and only if e is not contained in any chain in C. If G is 2-edge-connected, C is an ear decomposition. G is 2-vertex-connected if and only if G has minimum degree 2 and C 1 is the only cycle in C. A vertex v in a 2-edge-connected graph G is a cut vertex if and only if v is the first vertex of a cycle in C - C 1.
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The edge e belongs to a certain tree T⊆Fi. After the deletion of e, the tree T is broken to two smaller trees: Tx which contains the node x and Ty which contains the node y. An edge of Gi is a replacement edge, if and only if it connects a node in Tx with a node in Ty.
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