# The Median Problem on k-Partite Graphs

• Volume: 35, Issue: 3, page 439-446
• ISSN: 2083-5892

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## Abstract

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In a connected graph G, the status of a vertex is the sum of the distances of that vertex to each of the other vertices in G. The subgraph induced by the vertices of minimum (maximum) status in G is called the median (anti-median) of G. The median problem of graphs is closely related to the optimization problems involving the placement of network servers, the core of the entire networks. Bipartite graphs play a significant role in designing very large interconnection networks. In this paper, we answer a problem on the structure of medians of bipartite graphs by showing that any bipartite graph is the median (or anti-median) of another bipartite graph. Also, with a different construction, we show that the similar results hold for k-partite graphs, k ≥ 3. In addition, we provide constructions to embed another graph as center in both bipartite and k-partite cases. Since any graph is a k-partite graph, for some k, these constructions can be applied in general

## How to cite

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Karuvachery Pravas, and Ambat Vijayakumar. "The Median Problem on k-Partite Graphs." Discussiones Mathematicae Graph Theory 35.3 (2015): 439-446. <http://eudml.org/doc/271223>.

@article{KaruvacheryPravas2015,
abstract = {In a connected graph G, the status of a vertex is the sum of the distances of that vertex to each of the other vertices in G. The subgraph induced by the vertices of minimum (maximum) status in G is called the median (anti-median) of G. The median problem of graphs is closely related to the optimization problems involving the placement of network servers, the core of the entire networks. Bipartite graphs play a significant role in designing very large interconnection networks. In this paper, we answer a problem on the structure of medians of bipartite graphs by showing that any bipartite graph is the median (or anti-median) of another bipartite graph. Also, with a different construction, we show that the similar results hold for k-partite graphs, k ≥ 3. In addition, we provide constructions to embed another graph as center in both bipartite and k-partite cases. Since any graph is a k-partite graph, for some k, these constructions can be applied in general},
author = {Karuvachery Pravas, Ambat Vijayakumar},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {networks; distance; median; bipartite; k-partite.; -partite},
language = {eng},
number = {3},
pages = {439-446},
title = {The Median Problem on k-Partite Graphs},
url = {http://eudml.org/doc/271223},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Karuvachery Pravas
AU - Ambat Vijayakumar
TI - The Median Problem on k-Partite Graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 3
SP - 439
EP - 446
AB - In a connected graph G, the status of a vertex is the sum of the distances of that vertex to each of the other vertices in G. The subgraph induced by the vertices of minimum (maximum) status in G is called the median (anti-median) of G. The median problem of graphs is closely related to the optimization problems involving the placement of network servers, the core of the entire networks. Bipartite graphs play a significant role in designing very large interconnection networks. In this paper, we answer a problem on the structure of medians of bipartite graphs by showing that any bipartite graph is the median (or anti-median) of another bipartite graph. Also, with a different construction, we show that the similar results hold for k-partite graphs, k ≥ 3. In addition, we provide constructions to embed another graph as center in both bipartite and k-partite cases. Since any graph is a k-partite graph, for some k, these constructions can be applied in general
LA - eng
KW - networks; distance; median; bipartite; k-partite.; -partite
UR - http://eudml.org/doc/271223
ER -

## References

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5. [5] K. Pravas and A. Vijayakumar, Convex median and anti-median at pre- scribed distance, communicated.
6. [6] P.J. Slater, Medians of arbitrary graphs, J. Graph Theory 4 (1980) 389-392. doi:10.1002/jgt.3190040408[Crossref] Zbl0446.05029
7. [7] S.B. Rao and A.Vijayakumar, On the median and the anti-median of a co- graph, Int. J. Pure Appl. Math. 46 (2008) 703-710. Zbl1171.05014
8. [8] H.G. Yeh and G.J. Chang, Centers and medians of distance-hereditary graphs, Discrete Math. 265 (2003) 297-310. doi:10.1016/S0012-365X(02)00630-1 [Crossref] Zbl1026.05035

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