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By Cai M.-C., Favaron O., Li H.

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Example text

N in three different meanings. As names of vertices, as integers, and as letters of an alphabet. 4 The number of spanning trees for n distinct vertices is nn−2 . The remainder of this section describes a proof due to Prüfer [4]. ) Assume V = {1, 2, . . , n}. Let us display a bijection between the set of the spanning trees and the nn−2 words of length n − 2 over the alphabet {1, 2, . . , n}. 2. 2. We now apply TREEtoWORD. We start with the given T and an empty template for a word w of 4 letters.

Thus, T is the unique every tree that produces w must have an edge v tree that produces w. 4 Directed Tree Definitions A digraph G(V, E) is said to have a root r if r ∈ V and every vertex v ∈ V is reachable from r; that is, there is a directed path that starts in r and ends in v. A digraph (finite or infinite) is called a directed tree if it has a root and its underlying undirected graph is a tree. 5 Assume G is a digraph. The following five conditions are equivalent: (a) (b) (c) (d) G is a directed tree.

2 The Hopcroft-Tarjan Version of DFS The Hopcroft and Tarjan version of DFS is essentially the same as Trémaux’s, except that they number the vertices from 1 to n(= |V|) in the order in which they are discovered. This is not necessary, as we have seen, for scanning the graph, but the numbering is useful in applying the algorithm for more advanced tasks. Let us denote the number assigned to vertex v by k(v). Also, instead of marking passages, they mark edges as “used,” and instead of using the F mark to indicate the edge through which the vertex was discovered and through which it is left for the last time, let us record for each vertex v other than s the vertex f(v) from which v has been discovered.

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