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Problem 1:

public boolean hasSelfCycle( Graphnode<T> node ) {
    return hasSelfCycle(node, node);
}

private boolean hasSelfCycle(Graphnode<T> node, Graphnode<T> goal) {
    
    node.setVisitMark(true);

    for (Graphnode<T> succ : node.getSuccessors()) {

        if (succ == goal)  // if we've reached the first node
            return true;   // there is a self cycle

        else if (!succ.getVisitMark()) {
            if (hasSelfCycle(succ, goal))
                return true;
        }

    }
    return false;
}

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Problem 2:

Visited Vertexes and their         Priority Queue's items
shortest distances from start         (listed in increasing order)
-                                  ( 0, S)
S:0                                ( 4, G), (11, H), (33, P)
S:0, G:4                           (10, R), (11, H), (33, P)
S:0, G:4, R:10                     (11, H), (24, P), (30, A)
S:0, G:4, R:10, H:11               (12, P), (30, A)
S:0, G:4, R:10, H:11, P:12         (14, A)
S:0, G:4, R:10, H:11, P:12, A:14   empty

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Problem 3:

Possible topological orderings (only 2 required):
A C D B E F
A C D E B F
C A D B E F
C A D E B F
C D A B E F
C D A E B F