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On-line ranking of split graphs

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-ɛ)-competitive semi on-line algorithm of this type. Finally, a 2-competitive semi on-line algorithm is given.
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Piotr Borowiecki, Dariusz Dereniowski. On-line ranking of split graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2013, Vol. 15 no. 2 (2), pp.195--214. ⟨hal-00980767⟩

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