The median of a graph $G$ with weighted vertices is the set of all vertices $x$ minimizing the sum of weighted distances from $x$ to the vertices of $G$. For any integer $p\ge 2$, we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the $p$th power $G^p$ of $G$. This extends some characterizations of graphs with connected medians (case $p=1$) provided by Bandelt and Chepoi (2002). The characteristic conditions can be tested in polynomial time for any $p$. We also show that several important classes of graphs in metric graph theory, including bridged graphs (and thus chordal graphs), graphs with convex balls, bucolic graphs, and bipartite absolute retracts, have $G^2$-connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the isometric subgraphs of Johnson graphs and of halved-cubes with connected medians.

翻译：具有加权脊椎的Gog $G美元的中位值是将美元至G$的加权距离总和从美元到G$的加权距离的一组G$x美元。对于任何整数 $p\ge 2美元,我们用图表来描述图表,其中,对于任何非负加权,中位数总是引致以美元发电量($G$p美元)为单位的连接子图。这延伸了Bandelt和Chepoi(2002年)提供的与中位(美元=1美元)连接的中位数(案件=1美元)的图形的一些特征特征。典型条件可以在任何美元的多边时间测试。对于任何整数个整数 $p\ ge 2 美元,我们还显示,在图理中,包括连接的图形(因此是圆形图 )、 圆形球形球图、 泡球图和 双面绝对回调绝对回调的图表等若干重要的图表类别,都有以G+2美元连接的中位数。扩展了Bandelt 和 Chepoi 基的图表的结果是与中位数字的图的图表与中位图相连接的中位图。