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By Edsger W. Dijkstra, W. H. J. Feijen, Joke Sterringa

ISBN-10: 0201175363

ISBN-13: 9780201175363

Publication through Dijkstra, Edsger W., Feijen, W. H. J., Sterringa, shaggy dog story

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C) G est connexe et a n − 1 arˆetes. (d) G est un graphe connexe minimal (chaque arˆete est un pont). (e) G est un graphe minimal v´erifiant la propri´et´e δ(X) = ∅ pour tout ∅ = X ⊂ V (G). (f) G est un graphe maximal sans cycles (l’addition d’une arˆete quelconque cr´ee un cycle). (g) Toute paire de sommets de G est connect´ee par une chaˆıne unique. Preuve. (a)⇒(g), car l’union de deux chaˆınes distinctes ayant les mˆemes extr´emit´es contient un cycle. 3(a). (d)⇒(f) : e´ vident. (f)⇒(b)⇒(c) : cela se d´eduit du fait que si une forˆet a n sommets, m arˆetes et p composantes connexes, alors n = m + p.

Dans BFS nous choisissons pour v ∈ Q le premier sommet entr´e dans Q. Ici Q est impl´ement´e suivant une pile FIFO (first-in-first-out, en franc¸ais premier arriv´e, premier sorti). Un algorithme semblable a` DFS a e´ t´e propos´e avant 1900 par Tr´emaux et Tarry ; voir K¨onig [1936]. Il semble que l’expression BFS ait e´ t´e utilis´ee pour la premi`ere fois par Moore [1959]. Les arbres (dans le cas orient´e : les arborescences) (R, T ) calcul´es par DFS et BFS sont respectivement appel´es arbres-DFS et arbres-BFS.

4 Soit v1 , e1 , v2 , e2 , . . , vk , ek , vk+1 la suite W . For i := 1 to k do : Wi := E ULER(G, vi ). Set W := W1 , e1 , W2 , e2 , . . , Wk , ek , vk+1 . Retourner W . 5 Pour les graphes orient´es, remplacer 2 par : 2. Graphes 2 33 If δ + (x) = ∅ then go to 4 . Else soit e ∈ δ + (x) ; on peut supposer que e = (x, y). 25. L’ ALGORITHME D ’E ULER r´epond correctement. Il s’ex´ecute en un temps O(m + n) (n = |V (G)|, m = |E(G)|). Preuve. Par induction sur |E(G)|, le cas E(G) = ∅ e´ tant trivial.

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A Method of Programming by Edsger W. Dijkstra, W. H. J. Feijen, Joke Sterringa

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