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einen einfachen Weg der Länge $k-2$ in $G$ konstruieren, + indem wir $s$ und $t$ entfernen. + \points{2} + \item + Ein Hamiltonweg ist ein Weg der alle Knoten in $G$ beinhaltet + und somit Länge $n-1$ besitzt. + + Wie wir oben gezeigt haben, kann ein $s$-$t$-Weg der Länge $n+1$ + in $G'$ leicht in einen Weg der Länge $n-1$ in $G$ umgewandelt + werden. Das heißt, dass wir einen Hamiltonweg in $G$ finden, + wenn wir einen $s$-$t$-Weg finden. + + Umgekehrt können wir einen Hamiltonweg leicht in einen $s$-$t$-Weg + umwandeln, also finden wir einen $s$-$t$-Weg wenn wir einen + Hamiltonweg finden. + + Also finden wir einen Hamiltonweg genau dann, wenn wir einen + $s$-$t$-Weg finden. + \points{1} + \item + Da wir Hamiltonweg auf \algt{Längster $s$-$t$-Weg} reduziert + haben, muss also \algt{Längster $s$-$t$-Weg} $NP$-schwer sein, denn + wenn es in $P$ liegen würde, könnten wir auch Hamiltonweg in + polynomieller Zeit lösen. Da wir nicht von $P = NP$ ausgehen, + ist das nicht möglich. + \points{2} + \end{tasks}