@MISC{Borowiecki_on-lineranking, author = {Piotr Borowiecki and Dariusz Dereniowski}, title = {On-line Ranking of Split Graphs}, year = {} }

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Abstract

A vertex ranking of a graph G is an assignment of positive integers (colors) to the vertices of G such that each path connecting two vertices of the same color contains a vertex of a higher color. Our main goal is to find a vertex ranking using as few colors as possible. Considering on-line algorithms for vertex ranking of split graphs, we prove that the worst case ratio of the number of colors used by any on-line ranking algorithm and the number of colors used in an optimal off-line solution may be arbitrarily large. This negative result motivates us to investigate semi on-line algorithms, where a split graph is presented on-line but its clique number is given in advance. We prove that there does not exist a (2−ε)-competitive semi on-line algorithm of this type. Finally, a 2-competitive semi on-line algorithm is given.