NP-Completeness of the minimum edge-ranking spanning tree problem on series-parallel graphs

Arefin, Ahmed Shamsul; Mia, Md. Abul Kashem

doi:10.1109/iccitechn.2007.4579371

Cited by 3 publications

(2 citation statements)

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“…There exist exact polynomial-time algorithms solving the MERST problem for threshold and split graphs [10]. The problem turns out to be NP-complete for series-parallel graphs [1]. In [9] an approximation algorithm for series-parallel graphs is given.…”

Section: Introductionmentioning

confidence: 99%

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

Dereniowski¹

2009

Discuss. Math. Graph Theory

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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 NPhardness 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.

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Section: Introductionmentioning

confidence: 99%

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

Dereniowski¹

2009

Discuss. Math. Graph Theory

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“…Makino, Uno and Ibaraki introduced the minimum edge-ranking spanning tree problem (MERSTP) [9], that is, the problem of finding a spanning tree of a graph whose edge-ranking is minimum. MERSTP is also NP-hard [9], and the problem is still NP-hard even on series-parallel graphs [10]. An instance of the minimum spanning tree problem with label selection is illustrated in Fig.…”

Section: Introductionmentioning

confidence: 99%

Minimum Spanning Tree Problem with Label Selection

Fujiyoshi

Suzuki

2011

IEICE Trans. Inf. & Syst.

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SUMMARYIn this paper, we study the minimum spanning tree problem with label selection, that is, the problem of finding a minimum spanning tree of a vertex-labeled graph where the weight of each edge may vary depending on the selection of labels of vertices at both ends. The problem is especially important as the application to mathematical OCR. It is shown that the problem is NP-hard. However, for the application to mathematical OCR, it is sufficient to deal with only graphs with small tree-width. In this paper, a linear-time algorithm for series-parallel graphs is presented. Since the minimum spanning tree problem with label selection is closely related to the generalized minimum spanning tree problem, their relation is discussed.

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