The D’Esopo-Pape algorithm is exponential in the worst case. Consider the following method to construct pathological graphs of n vertices (Kershenbaum 1981), each vertex identified by a number 1, . . . , n:
KershenbaumAlgorithm() construct a two-vertex graph with vertices 1 and 2, and edge(1,2) = 1; for k = 3 to n add vertex k; for i = 2 to k21 add edge(k,i) with weight(edge(k,i)) = weight(edge(l,i)); weight(edge(l,i)) = weight(l,i) + 2k–3 + 1; add edge(l,k) with weight(edge(l,k)) = 1;
The vertices adjacent to vertex 1 are put in ascending order and the remaining adjacency lists are in descending order. Using this algorithm, construct a five-vertex graph and execute the D’Esopo-Pape algorithm showing all changes in the deque and all edge updates. What generalization can you make about applying Pape’s method to such graphs?
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