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

• Volume: 29, Issue: 2, page 253-261
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Dariusz Dereniowski. "Minimum vertex ranking spanning tree problem for chordal and proper interval graphs." Discussiones Mathematicae Graph Theory 29.2 (2009): 253-261. <http://eudml.org/doc/270173>.

@article{DariuszDereniowski2009,
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.},
author = {Dariusz Dereniowski},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {computational complexity; vertex ranking; spanning tree},
language = {eng},
number = {2},
pages = {253-261},
title = {Minimum vertex ranking spanning tree problem for chordal and proper interval graphs},
url = {http://eudml.org/doc/270173},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Dariusz Dereniowski
TI - Minimum vertex ranking spanning tree problem for chordal and proper interval graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2009
VL - 29
IS - 2
SP - 253
EP - 261
AB - 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.
LA - eng
KW - computational complexity; vertex ranking; spanning tree
UR - http://eudml.org/doc/270173
ER -

## References

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