H. Bodlaender, J.S. Deogun, K. Jansen, T. Kloks, D. Kratsch, H. Muller, Z. Tuza, Rankings of graphs, SIAM J. Discrete Math. 11 (1998) 168-181. Home/Search Document Details and Download
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Optimal Edge Ranking of Trees in Linear Time - Lam, Yue (1998) (3 citations) (Correct)
....an intermediate edge with label j i. Figure 1 illustrates an optimal edge ranking. 2 1 2 1 3 4 1 1 3 5 4 1 2 3 Figure 1: An optimal edge ranking of a tree. The node ranking and edge ranking problems have been studied by a number of researchers as they find applications in different context [1, 6, 10, 11, 14]. Both problems are now known to be NP hard for graphs [12, 9, 7] Nonetheless, in most applications, the graphs in concern are restricted to trees only. This initiates the study of node ranking and edge ranking of trees. With respect to trees, the node ranking problem seems easier than the edge ....
....are generated. At the end of each iteration, the labels stored in the array B define a partial labeling. The positive labels in use always form a subset of the labels used by an optimal labeling (see Section 4) That means, the procedure never uses a wrong label. On exit from the while loop, B[1]; B[2] B[d] are all positive and form a valid labeling. In Section 4 we give a characterization of this valid labeling and prove that it is actually optimal. Based on the procedure labeling, we can construct an algorithm for computing a critical edge ranking of a rooted tree R. For each ....
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H. L. Bodlaender, J. S. Deogun, K. Jansen, T. Kloks, D. Kratsch, H. Muller, and Zs. Tuza, Rankings of graphs, SIAM J. Discrete Math. To appear.
On the Vertex Ranking Problem for d-trapezoid Graphs.. - Deogun, Kloks.. (1999) Self-citation (Deogun Kloks Kratsch) (Correct)
.... an O(n 4 ) algorithm for vertex ranking of interval graphs was presented in [1] The vertex ranking problem Given a graph G and a positive integer k , does G have a vertex ranking with at most k colours is NP complete, even when restricted to cobipartite and bipartite graphs, respectively [2]. In view of this, it is interesting to notice that for each constant t , the class of graphs with vertex ranking number at most t is minor closed. Since this class of graphs clearly does not contain all planar graphs, for example in [15] it is proved that the n Theta n grid has vertex ranking ....
....number at most t is minor closed. Since this class of graphs clearly does not contain all planar graphs, for example in [15] it is proved that the n Theta n grid has vertex ranking number at least n) it follows that graphs with bounded vertex ranking number are recognizable in linear time [2]. In [15] among other things, an O( p n) bound is given for the vertex ranking number of a planar graph and the authors describe a polynomial time algorithm, which finds a ranking of a given planar graph using only O( p n) colours. For graphs in general there is an approximation algorithm ....
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H. Bodlaender, J.S. Deogun, K. Jansen, T. Kloks, D. Kratsch, H. Muller and Z. Tuza, Rankings of graphs, Proceedings of the 20th International Workshop on Graph-Theoretic Concepts in Computer Science, SpringerVerlag, Lecture Notes in Computer Science 903, 1995, pp. 292--304.
Vertex Rankings of Chordal Graphs and Weighted Trees - Dereniowski, Nadolski (2006) (Correct)
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H. Bodlaender, J.S. Deogun, K. Jansen, T. Kloks, D. Kratsch, H. Muller, Z. Tuza, Rankings of graphs, SIAM J. Discrete Math. 11 (1998) 168-181.
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