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Published in:
05-05-2016
Convex median and anti-median at prescribed distance
Published in: Journal of Combinatorial Optimization | Issue 3/2017
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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 \(H=(G_1,G_2,r)\) denote a graph with \(G_1\) as the median and \(G_2\) as the anti-median of H, \(d(G_1,G_2)=r\) and both \(G_1\) and \(G_2\) are convex subgraphs of H. It is known that \((G_1,G_2,r)\) exists for every \(G_1\), \(G_2\) with \(r \ge \left\lfloor diam(G_1)/2\right\rfloor +\left\lfloor diam(G_2)/2\right\rfloor +2\). In this paper we show the existence of \((G_1,G_2,r)\) for every \(G_1\), \(G_2\) and \(r \ge 1\). We also obtain a sharp upper bound for the maximum status difference in a graph G.