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By Rudenskaya O. G.

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Ek , xk = x where xi = xj with 0 ≤ i < j ≤ k. The walk can be shortened by removing the subsequence (subwalk) between xi and xj , which gives a new walk still linking x and x : x = x0 , e1 , x1 , . . , xi = xj , . . , ek , xk = x By repeating this shortening process as long as there is a vertex that can be found twice in the walk, that is as long as the walk is not a path, we end up obtaining a path linking the vertices x and x . 2 Cycles A walk, a trail or a path (x0 , e1 , x1 , . . , ek , xk ) is said to be closed if its ends x0 and xk coincide.

Remove, as long it is possible, an edge which is not a bridge (first in graph G, and then in the current graph obtained). The spanning subgraph G obtained is connected, like G, because each of the edges removed was not a bridge. 1). This graph G is therefore a tree, spanning a subgraph of G. Let m be the number of edges of G . We have m = n − 1 = m. Thus, G having the same number of edges as G, G = G and G is therefore a tree. Note. The method of proving equivalences between some conditions, which is used here to prove this theorem, is classic.

A second principle for implementing a graph is to give for each vertex its “neighborhood”: the vertices which are its neighbors or its incident edges or even both, that is its incident edges and their endvertices. This last method will be particularly useful when there are multiple edges. Implementing this can be done in various ways, the most classic, from a programming point of view, is to give, for example, the neighbors in a list called an adjacency list or list of neighbors. If we want to modify the graph during its processing, it is best to implement these lists under the classic form of linked lists.

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3-Quasiperiodic functions on graphs and hypergraphs by Rudenskaya O. G.

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