Convex median and anti-median at prescribed distance

K. Pravas () and A. Vijayakumar
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K. Pravas: KKTM Government College
A. Vijayakumar: Cochin University of Science and Technology

Journal of Combinatorial Optimization, 2017, vol. 33, issue 3, No 14, 1029 pages

Abstract: Abstract The status of a vertex v in a connected graph G is the sum of the distances between v and all the other vertices of G. The subgraph induced by the vertices of minimum (maximum) status in G is called median (anti-median) of G. Let $$H=(G_1,G_2,r)$$ H = ( G 1 , G 2 , r ) denote a graph with $$G_1$$ G 1 as the median and $$G_2$$ G 2 as the anti-median of H, $$d(G_1,G_2)=r$$ d ( G 1 , G 2 ) = r and both $$G_1$$ G 1 and $$G_2$$ G 2 are convex subgraphs of H. It is known that $$(G_1,G_2,r)$$ ( G 1 , G 2 , r ) exists for every $$G_1$$ G 1 , $$G_2$$ G 2 with $$r \ge \left\lfloor diam(G_1)/2\right\rfloor +\left\lfloor diam(G_2)/2\right\rfloor +2$$ r ≥ d i a m ( G 1 ) / 2 + d i a m ( G 2 ) / 2 + 2 . In this paper we show the existence of $$(G_1,G_2,r)$$ ( G 1 , G 2 , r ) for every $$G_1$$ G 1 , $$G_2$$ G 2 and $$r \ge 1$$ r ≥ 1 . We also obtain a sharp upper bound for the maximum status difference in a graph G.

Keywords: Networks; Distance; Median; Facility location; Convex (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-016-0022-z

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