Discussiones Mathematicae Graph Theory 29(2) (2009) 253-261
doi: 10.7151/dmgt.1445

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MINIMUM VERTEX RANKING SPANNING TREE PROBLEM FOR CHORDAL AND PROPER INTERVAL GRAPHS

Dariusz Dereniowski

Department of Algorithms and System Modeling
Gdańsk University of Technology, Poland
e-mail: deren@eti.pg.gda.pl

Abstract

A vertex k-ranking of a simple graph is a coloring of its vertices with k colors in such a way that each path connecting two vertices of the same color contains a vertex with a bigger color. Consider the minimum vertex ranking spanning tree (MVRST) problem where the goal is to find a spanning tree of a given graph G which has a vertex ranking using the minimal number of colors over vertex rankings of all spanning trees of G. K. Miyata et al. proved in [NP-hardness proof and an approximation algorithm for the minimum vertex ranking spanning tree problem, Discrete Appl. Math. 154 (2006) 2402-2410] that the decision problem: given a simple graph G, decide whether there exists a spanning tree T of G such that T has a vertex 4-ranking, is NP-complete. In this paper we improve this result by proving NP-hardness of finding for a given chordal graph its spanning tree having vertex 3-ranking. This bound is the best possible. On the other hand we prove that MVRST problem can be solved in linear time for proper interval graphs.

Keywords: computational complexity, vertex ranking, spanning tree.

2000 Mathematics Subject Classification: 68R10, 68Q25, 05C15.

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Received 3 December 2007
Revised 10 March 2009
Accepted 10 March 2009