Vertex rankings of chordal graphs and weighted trees

Authors:
Dariusz Dereniowski;Adam Nadolski
Affiliations:
Department of Algorithms and System Modeling, Gdańsk University of Technology, Poland;Department of Algorithms and System Modeling, Gdańsk University of Technology, Poland
Venue:
Information Processing Letters
Year:
2006

Citing 6
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Abstract

In this paper we consider the vertex ranking problem of weighted trees. We show that this problem is strongly NP-hard. We also give a polynomial-time reduction from the problem of vertex ranking of weighted trees to the vertex ranking of (simple) chordal graphs, which proves that the latter problem is NP-hard. In this way we solve an open problem of Aspvall and Heggernes. We use this reduction and the algorithm of Bodlaender et al.'s for vertex ranking of partial k-trees to give an exact polynomial-time algorithm for vertex ranking of a tree with bounded and integer valued weight functions. This algorithm serves as a procedure in designing a PTAS for weighted vertex ranking problem of trees with bounded weight functions.