MEDIANS OF GRAPHS AND KINGS OF TOURNAMENTS*

Abstract

We first prove that for any graph $G$ with a positive vertex weight function $w$, there exists a graph $H$ with a positive weight function $w'$ such that $w(v)=w'(v)$ for all vertices $v$ in $G$ and whose $w'$-median is $G$. This is a generalization of a previous result for the case in which all weights are 1. The second result is that for any $n$-tournament $T$ without transmitters, there exists an integer $m\leq 2n-1$ and an $m$-tournament $T'$ whose kings are exactly the vertices of $T$. This improves upon a previous result for $m\leq 2n$.