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ArticleOriginal scientific text

Title

Minimum vertex ranking spanning tree problem for chordal and proper interval graphs

Authors 1

Affiliations

  1. Department of Algorithms and System Modeling, Gdańsk University of Technology, Poland

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

ENG
computational complexity, vertex ranking, spanning tree

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Discussiones Mathematicae Graph Theory
2009/29/2
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Pages:
253-261
Main language of publication
English
Received
2007-12-03
Accepted
2009-03-10
Published
2009

DOI: 10.7151/dmgt.1445

Exact and natural sciences