Convex median and anti-median at prescribed distance

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.