Abstract
Graphs and Algorithms 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.
Highlights
In this paper we consider the vertex ranking problem for simple, finite, undirected graphs G = (V, E) with the vertex set V, edge set E and order n = |V (G)|
From the results proved in this paper it follows that for split graphs there does not exist a constant competitive on-line ranking algorithm that could guarantee the usage of an interval of colors containing color 1
In the theorem we prove that for any ε > 0, there does not exist a semi on-line algorithm A that is (2 − ε)-competitive, even when A is given the clique number of a split graph G and colors v1, v2 using colors greater than ω(G)
Summary
The remaining parts of the paper focus on the statement and analysis of a 2-competitive semi on-line ranking algorithm that knows the clique number of a split graph in advance. From the results proved in this paper it follows that for split graphs there does not exist a constant competitive on-line ranking algorithm that could guarantee the usage of an interval of colors containing color 1.
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