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
@article{bwmeta1.element.doi-10_7151_dmgt_1802,
author = {Karuvachery Pravas and Ambat Vijayakumar},
title = {The Median Problem on k-Partite Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {35},
year = {2015},
pages = {439-446},
zbl = {1317.05043},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1802}
}
Karuvachery Pravas; Ambat Vijayakumar. The Median Problem on k-Partite Graphs. Discussiones Mathematicae Graph Theory, Tome 35 (2015) pp. 439-446. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1802/
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